intro to astro
homework 5
This homework is a bit more calculation-oriented than average; homework
6 will be more balanced between qualitative and quantitative questions.
1) This question will explore energy balance in the supernova process
that we talked about on Wednesday, August 2.
We'll use Supernova 1987A as an example. Let's assume that the
precursor star was 20 solar masses (p. 501). Further assume it's core
collapsed to form a neutron star of 2 solar masses of radius 10 km; the
remainder was blown off in the explosion.
a) gravitational
energy:
(1) to derive the formula for the energy released in the gravitational
collapse of a star of mass M is bit beyond the level of our course, so
let's do something simpler that gives us virtually the same answer:
dimensional analysis. I think you would agree that the
gravitational energy released in an implosion of mass M to a final radius R depends on M, R, and the
gravitation constant G (which
measures the strength of gravity in the universe). There is only
one combination of G, M, and R that has units of energy. To find
it, let's assume
grav energy released
= Gx My Rz
where x, y, and z are unknown powers to be found by matching the units
on both sides. Plug in the various units for each quantity
(energy, G, M, and R), making sure that you use only basic units (m,
kg, s); then collect units on each side; then match the exponent on the
left side to the exponent on the right side in turn for each of the
three basic units to solve for x, y, and z
(2) Now that you have the formula for gravitational energy in terms of
G, M, and R, you can calculate the energy released in joules during the
supernova implosion. It should compare well with the number in
the book. [As I'm sure you realize, the method of dimensional
analysis cannot determine multiplying dimensionless such as 2p or 1/2 or .... It is amazing
(or maybe it's just divine guidance) that these pure numerical
multiplying constants are always about 1. The multiplying factor
that appears in a more physical calculation here is 0.6 .]
b) neutrino energy:
(1) first calculate the number of neutrons in the (completely-neutron)
neutron star.
(2) since you know how the number of neutrons produced during the
supernova implosion related to the number of neutrinos produced (page
499), you now know the latter also
(3) the book gives an average neutrino energy on p. 503 and tells how
this was measured. Use this average energy to calculate the total
energy released in the form of neutrinos. It should match the
number given in the book. (Actually this calculation
underestimates the number of neutrinos by a factor of about 5; during
the implosion a significant amount of photon pairs are converted to
neutrino pairs, thereby increasing the number of neutrinos.)
c) kinetic energy:
on a previous homework, you calculated [text, problem 20(39)] the speed
of the shock wave associated with a supernova.
However, this calculation was the average speed of the shock wave over
the 10,000 years of expansion. Clearly the shock wave loses speed
as it expands. Let's assume that the real ejection speed of the
supernova matter was 4x higher than the speed calculated in 20(39).
Find the total kinetic energy of the ejecta.
d) light energy:
another non-trvial calculation, since we have an exponetially decaying
luminosity (remember the radioactive decay graph I showed you for SN
1987A in class?) that would need to be integrated over all time, but
once again dimensional analysis comes to the rescue. What could
the light energy depend on? simply the maximum luminosity
(energy/time) and the half-life of the decay (a time). The book
tells you the initial luminosity (page 501, found, as you know, from
the distance to SN 1987A and its apparent magnitude). You may
remember from class that the main radioactive product, cobalt-56, has a
half-life of 77 days. Use the max luminosity and the cobalt
half-life to calculate the total light energy emitted.
e) rotational
kinetic energy:
(1) We presently have no evidence for the presence (or absence) of a
neutron star left behind by SN 1987A. (One of the test 2
questions addressed this issue.) So let's assume that it has the
same (rotational) period as the most recently formed pulsar: the one in
the Crab Nebula. Use this period (page 515) and the radius, to
find the equatorial speed of rotation of our putative neutron star.
(2) Use this speed in the usual formula for KE to find the rotational
kinetic energy of the neutron star.
2) In this problem we prove that the pulsar phenomenon is due to
neither a pulsating white dwarf nor a pulsating neutron star.
a) How fast a star can expand or contract due to pulsations depends (as
did the gravitational energy above) on only three things: G, M, and R.
Once again, the calculation of the natural pulsation period of a star
is a bit involved, requiring the solving of a differential
equation. Instead, we'll
do a new dimensional analysis (like the one done in part a above) to
determine x, y, and z so that the combination gives units of time.
Find x,y, and z that gives units of time.
b) Pick a typical mass and radius for a neutron star (well, that's
easy, we just used such a pair in the previous problem), and calculate
the period of such an object.
Does it fit the observed range of pulsar periods (.01 sec - 2 sec)?
c) Repeat the calculation for a typical white dwarf. Does this
period fit the observed pulsar?
3)
problem 24(35), which accounts for how we know
that the pulsar phenomenon is not due to an eclipsing binary with two
neutron stars. (on test 2, question 1, you could easily
rule out such a binary containing at least 1 white dwarf, but you
couldnt rule out a binary containing two neutron stars. The
number you
calculate here is actually an upper limit to the age of the system,
because the rate at which the separation of the two stars increases
with
time as the stars approach each other and the gravitational effects
increase.)
this question is now number 1 on
homework 6