Astr 221 Homework #6 -- Due Nov 18
1. Visual Binaries
Sirius is a visual binary with a
of 49.94 years. It has a parallax of 0.377 arcseconds, and the
axis of the true relative orbit is 7.62 arcseconds. The ratio of the
distances of Sirius A and Sirius B from the center of mass is
a(A)/a(B)=0.466. Sirius A has an apparent bolometric magnitude of
while Sirius B has an apparent bolometric magnitude of 5.69.
- Find the mass of each star (in solar masses)
- Find the luminosity of each star (in solar luminosities)
- If the temperature of Sirius B is 27,000 K, estimate its
size and compare it to the size of the Earth.
- Calculate the density of Sirius B. What is the mass
of a teaspoonful of Sirius B?
2. Recreating the Main
- Using the Stellar
Structure Java Applet, construct models of stars of solar
metallicity (X=0.7, Y=0.28, Z=0.02), with the following masses (in
- Plot these models on a theoretical
Hertzprung-Russell Diagram. Make sure your axes are correct - y
should be Log L increasing upwards, x should be log Teff
increasing to the left!
- Plot log(L) versus log(Mass), and then
derive the mass-luminosity relationship: L~Mx.
- What is the value for x that fits the whole
sequence of models?
- If you only consider low mass stars (say
M<2Msun), what is the value for x?
- If you only consider high mass stars (M>5Msun),
what is the value for x?
- Make plots of how core temerature (Tc)
and core density (rhoc) depend on stellar mass. (Again, make
these plots log-log plots.) Based on these plots, which is more
important for driving high nuclear reaction rates -- the temperature or
the density? Explain why.
3. Equation of State
- Make a (log-log) plot of log Tc
log rhoc for the models you calculated above. On that plot,
draw a line showing where radiation pressure is equal to the ideal gas
law pressure, and then mark which regions of the plot are dominated by
radiation pressure and which are dominated by gas pressure. Are stars
dominated by radiation pressure or gas pressure in their centers?
4. The Rigel Incident
On the way to the star cluster, the USS Asinus receives
a distress call from the direction of Rigel (Rigel is a blue
supergiant, with a luminosity of L=40,000 Lsun, and temperature
T=11,200K). The message reads as follows:
Calling USS Asinus. This is Agricultural
Outpost Nine, on a planet orbiting Rigel at a distance of 1 AU. Three
Hoarsflize battleships have entered our system and have taken up a
strategic position 10 AU away from us. We suspect attack is imminent.
Can you evacuate our people?
Now normally even the mighty Asinus would be no match
for three heavily armed Hoarsflize battleships. But on this mission we
are testing out a new cloaking device, the NoVu9000, which absorbs all
energy it receives. This is particularly useful against Hoarsflize
sensors, which use AEP (active electromagnetic probing) technology. The
only problem -- the NoVu9000 has been tested in deep space, but never
close to any stars.
The NoVu9000 works by setting up a cloaking field
with a 1 km radius around the Asinus. However, it is very sensitive to
strong radiation -- if it absorbs more than 5x1024 ergs of
energy it will fail.
Question 1: How
much time would we have to evacuate the Outpost before our cloaking
device would fail?
Even with the cloaking device blocking the
Hoarsflize's AEP sensors, it is still possible that they could make visual
with the Asinus (why?). The cameras onboard the Hoarsflize warships are
still fairly sensitive, able to detect objects down an apparent visual
magnitude of about mV = 9.5.
Question 2: Could
the Hoarsflize warships see us even if we were cloaked?
To evacuate the outpost will take 30 minutes, if we
can avoid the warships. If the warships see us, we're toast. And if we
don't save the outpost when we could have, we'll all be busted down to
privates by Starfleet.
Make a Decision: Rescue