Download Problem set #2: AY 254C (Spring 2014) Due March 3, 2014

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 ?