Download Prep Homework Solutions for HW due 10/04/10

Prep Homework Solutions for HW due 10/04/10
1) The Algol system seems odd because the lower-mass star seems to be the more
evolved star, but normally we think of binaries as stars born together and we expect
higher-mass stars to evolve faster. The resolution of the paradox is presumed to be that
the red giant in Algol used to be the more massive star, and it evolved off the Main
Sequence before its companion, but then it lost significant mass through mass transfer to
the companion, so the more massive star is the less massive star now.
Note: a couple of you suggested that the paradox could be resolved by supposing that
stars in binaries change each other rate of aging or “life cycles.” This idea is not really
correct – here the more massive star did evolve faster, it just wasn’t obvious.
2) A core-collapse supernova occurs when a massive star develops an iron core, which is
unable to fuse into higher elements via a reaction that produces energy. At that point the
star has no way to sustain outward pressure from the center and it implodes; the rebound
of the outer layers when they fall in due to gravity and “bounce” off the core generates
the explosion. A thermonuclear supernova occurs when mass transfer onto a white dwarf
in a close binary causes its mass to exceed the Chandrasekhar limit of 1.4 Msun, which is
the maximum mass possible for a white dwarf. Above this mass, gravitational
compression ignites fusion in the core, converting C and O to Nickel. The fusion blows
the white dwarf apart since it is not composed of “normal” matter that could expand and
cool to achieve hydrostatic equilibrium.
Thermonuclear supernovae differ from core-collapse supernovae in that they have a
completely different source of energy (nuclear fusion rather than gravitational collapse
energy) and do not represent the deaths of massive stars, but rather represent a stage of
evolution for aging binary systems. Because thermonuclear supernovae blow apart
completely, they do not leave a remnant neutron star and do not emit large amounts of
neutrinos. In contrast, core-collapse supernovae create vast amounts of neutrinos when all
their protons turn into neutrons as the core condenses into the remnant neutron star.
3) Calculate the orbital speed of gas on the inner edge of an accretion disk around a
neutron star of mass 1.4 solar masses and radius 10 km.
Rearranging the motions find mass equation and using the value of G from class 7,
V2 = GM/R = (4.28x10-6 kpc (km/s)2 Msun-1)(1.4 Msun) / ((10km)(3.24x10-17 kpc/km)).
Plugging in and taking the square root, V=1.36 x 105 km/s (a good fraction of the speed
of light, so actually our calculation would have benefitted from a relativistic treatment!).
- OVER -
4) What prevents thermonuclear reactions from occurring at the center of a white
dwarf? If no thermonuclear reactions are occurring in its core, why doesn't the star
collapse?
A white dwarf is composed primarily of C and O in a dense structure supported by
electron degeneracy pressure. (This is the quantum mechanical effect that two electrons
cannot occupy the same state. NOTE: it is not simply the fact that like charges repel!)
Despite their high density, white dwarfs have inadequate central pressure and temperature
to enable further fusion of C and O into higher elements. If a white dwarf accretes
enough mass to cross the Chandrasekhar limit of 1.4 Msun, then the pressure will
increase just enough to ignite fusion, igniting a runaway explosion (since the star is solid
and thus incapable of adjusting its internal pressure to achieve hydrostatic equilibrium).
5) What is neutron degeneracy pressure? How does it enable neutron stars to exceed
the Chandrasekhar limit?
Neutron degeneracy pressure is the resistance of neutrons to sharing the same state, due
to quantum mechanics. It can withstand gravity even when electron degeneracy pressure
fails – in fact the failure of e- degeneracy pressure leads to the formation of neutrons (as
atomic nuclei and electrons combine), so when a compact remnant exceeds the
Chandrasekhar limit, i.e., the mass limit for support by e- degeneracy pressure as in a
white dwarf, it can become a neutron star and withstand the increased gravity from its
higher mass in that form.
General Note: At least one of you was confused and thought thermonuclear fusion would
break apart bonds, fusing metals into simpler “basic atoms.” This is actually backward.
Fusion is the process that builds up nuclei from the simplest H atom to produce the
metals in the Universe. In a normal star’s core, the heaviest nucleus produced is Fe,
because after that the heavier nuclei cannot be formed in reactions that release energy,
but only in reactions that require input energy (which do occur in special situations such
as during supernova explosions).