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Problem set #2: AY 254C (Spring 2014) Due March 3, 2014 (5pm) 1. Heavy Metal Atmosphere Some white dwarfs have been found to have metals in their atmospheres. In this problem you will understand why this is strange. (a) What is the macroscopic force on an ion with charge Ze and mass Amu in a star otherwise composed of pure hydrogen? (Recall the electric field eE = mp g/2 that has to be present to keep the protons from sinking and the electrons from flying out of the star.) (b) Roughly how often does an ion collide with the electrons and protons in the gas? Which collisions determine the rate of downward settling? Estimate numerical values for ions in the atmosphere of an 0.6 M white dwarf. (c) Combining (a) and (b), calculate the rate of downward drift of an ion through the white-dwarf atmosphere as a function of the density, temperature, and gravity in the gas. How does the time to drift through the atmosphere scale with the temperature of the atmosphere (scale to a fiducial value of 104 K)? Should we see heavy elements in the WD atmosphere? or just pure hydrogen? 2. Brunt-Väisälä frequency White dwarfs are observed to undergo oscillations with periods (very roughly) of ∼500 sec. Read up on WD pulsations 1 Using typical white-dwarf numbers (e.g., carbon interior, mass M = 0.6 M and radius R = 0.01 R ), to show that the the BruntVäisälä frequency in the outer envelope of the white dwarf is roughly consistent with the observed g-mode frequency. How does this result depend on the internal temperature? 3. Neutron-star order-of-magnitude estimates. (a) Rotation period. Imagine our Sun collapses to a radius of 10 km. What would the final spin period be, assuming conservation of angular momentum and assuming the initial and final states are both homogeneous spheres. (b) Kick velocity. Assume that a newborn neutron star radiates 1053 erg in neutrinos. Show that an asymmetry in emission as small as 1% (for example, if the star radiates slightly more neutrinos in the positive-z direction than in the negative-z direction) is enough to impart a kick of several 100 km s−1 to the neutron star (assume a mass of 1.4 M ). Make use of the fact that neutrinos are ultrarelativistic particles. 1 Eg. M. Montgomery “White Dwarf and PreWhite Dwarf Pulsations” http://proceedings.aip.org/ resource/2/apcpcs/1170/1/605_1; D. Winget “Asteroseismology of white dwarf stars” http://adsabs. harvard.edu/abs/1998JPCM...1011247W. Bildsten & Cummings (1998). (c) Magnetic field. Assume a star of the size and magnetic field of the Sun collapses to a radius of 10 km. Estimate the magnetic field of the resulting object, assuming that magnetic flux is conserved. (d) Temperatures and accretion rates. Consider a binary system in which matter accreting onto a compact object leads to a luminosity close to 1038 erg/sec, close to the Eddington luminosity for a M star. Show that if the compact object is a white dwarf then it radiates in the ultraviolet, but if it is a neutron star, it radiates primarily in x-rays. Find the mass accretion rate onto a neutron star consistent with this luminosity. (e) Maximum magnetic fields. As the magnetic field in a neutron star is increased, the energy density in the magnetic field also increases. Estimate the magneticfield strength above which the magnetic-field energy density becomes larger than the (rest-mass) energy density in the outer crust of the neutron star. Estimate the magnetic-field strength above which the magnetic-field energy density becomes larger than the (rest-mass) energy density in the core of the neutron star. Deduce from these estimates a rough upper limit to the allowed value of the magnetic field in a magnetar. 4. Glitches aint glamourous (a) The spinup of the crust in a pulsar glitch is communicated to the charged particles in the interior of the neutron star by the magnetic fields that thread them. Any distortion in the magnetic field due, for example, to differential rotation, generates magnetic “sound waves,” or Alfvén waves, which travel with a speed vA ∼ (PB /ρ)1/2 , where PB ∼ B 2 /8π and ρ is the mass density. Estimate the time τA for the crust spinup to be communicated to the interior of the star. Find and use the magnetic-field strength typical for the Vela pulsar and compare (roughly) with the post-glitch healing time for the Vela pulsar. (b) Argue, using energetic considerations, that the magnetic field in the closed fieldline region about a pulsar can trap charged particles up to a maximum plasma moment of inertia, Bp2 R3 Ip ∼ . 6Ω2 Calculate the fractional angular-frequency change ∆ΩΩ if all of the plasma were released suddenly without generating a torque on the star. Evaluate your answer for the Crab and Vela pulsars, assuming reasonable values for Bp , M , and R. 5. Steady-State Burning of Pure Helium on a Neutron Star. This is a problem originally from Lars Bildsten (a former Berkeley prof, now in Santa Barbara). It covers the physics associated with Type-I X-ray bursts, which we did not cover in detail class. The neutron star in a low-mass x-ray binary accretes hydrogenheliumrich matter from a companion star. This accreted matter winds up getting spread over the surface of the neutron star. As additional layers of matter are accreted, those underneath are compressed and heated. There are several possibilities. For example, there may be slow steady-state burning of hydrogen-helium to heavier elements in these accreted layers. But if the conditions (e.g., accretion rate, composition of accreted matter) are right, the base layers of accreted matter may reach temperatures and densities at which the heating due to nuclear burning outpaces the rate at which the heat can be radiated away; the result is then a runaway thermonuclear explosion that appears to us as a Type-I x-ray burst. The purpose of this problem is to work through some of the relevant physics; you may wish to refer to Bildstens article (astro-ph/9709094; link from class bspace “Reading”) as you do this problem. Well consider accretion of pure helium onto a neutron star of mass M = 1.4M and R = 10 km. There are only a few such systems known in our galaxy, most of which burn unstably and give Type I X-ray bursts. We will consider only one reaction, which is helium burning to carbon, for which energy is generated at the rate, 3α = 5.3 × 1021 erg gm−1 s−1 ρ25 −44 exp 3 T8 T8 where ρ5 = ρ/105 gm cm−3 and T8 = T /108 K. The amount of energy released from 3α → 12 C is E3α = 5.841017 erg gm−1 . For the equation of state, presume an ideal gas of fully ionized alpha particles, so that P = (ne + n4 )kT , where ne = 2n4 is the number density of electrons and n4 is the number density of α particles. Lets find out where and when the helium burns after it enters the atmosphere. Work in plane-parallel coordinates, where ṁ is the accretion rate per unit surface area, in units gm cm−2 s−1 . (a) The flux leaving the atmosphere is set by heat transport F =− c d c d aT 4 = aT 4 3κρ dz 3κ dy where a is the radiation constant. The material is in hydrostatic balance, so dP/dz = −ρg, where g = GM/R2 is constant. Presume a constant flux and κ = 2mp /σTh = 0.2 cm2 gm−1 to show that this equation integrates to T4 = 3κP F acg at pressures large compared to the photospheric pressure. (b) If the fuel is burning in steady-state, then what is the flux leaving the top of the atmosphere as a function of ṁ? (c) Use the flux found in part (b) in the equation derived in part (a) to calculate the dependence of temperature on density and/or pressure in the neutron star atmosphere. (d) The helium burns when the lifetime of the helium nucleus to the 3α reaction (tburn = E3α /3α ) is comparable to the time it takes to get to that depth, taccr = P/g ṁ. Using the relations found previously, solve the resulting transcendental equation for the temperature where the burning occurs when ṁ = 105 gm cm−2 s−1 . What is the pressure at the depth of the helium burning? How long does it take for the matter to reach that depth? (HINT: In order to get some feeling for things, you can simplify the transcendental by expanding the exponential about a characteristic temperature of T8 = 4.) (e) Plot the burning temperature, column depth (P/g), taccr , and density as a function of ṁ for the range 104 gm cm−2 s−1 < ṁ < 106 gm cm−2 s−1 . Are the electrons always non-degenerate for these accretion rates? (f) If the gas is cool enough, T8 < 5, then the temperature dependence of the triplealpha reaction is steep enough that there will be explosive burning of helium. If, however, T8 > 5, the helium will burn stably (see, e.g., Section 2.5 in Bildstens article). What accretion rate does T8 > 5 require in units of the local Eddington rate? (g) Observationally, magnetic X-ray pulsars never show evidence for X-ray bursts. This is most likely because the local accretion rate at the polar caps exceeds the critical value found in (f) and that the burning occurs while the matter is still at the polar cap. This means that the accretion flow must only be over a small fraction of the neutron star area. What fraction is that for a neutron star with a total accretion luminosity of L = 1036 erg s−1 ?