Astr 221 Homework #6 -- Due Nov 18

1. Visual Binaries

Sirius is a visual binary with a period of 49.94 years. It has a parallax of 0.377 arcseconds, and the semimajor 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 -1.55, 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 Sequence

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 solar masses):

0.3	1.2	2.5	10.0
0.6	1.4	3.0	25.0
0.8	1.7	5.0	40.0
1.0	2.0	8.0	80.0

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 T_eff increasing to the left!

Plot log(L) versus log(Mass), and then derive the mass-luminosity relationship: L~M^x.

What is the value for x that fits the whole sequence of models?
If you only consider low mass stars (say M<2M_sun), what is the value for x?
If you only consider high mass stars (M>5M_sun), what is the value for x?

Make plots of how core temerature (T_c) and core density (rho_c) 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 T_c versus log rho_c 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 contact 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 m_V = 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 or not?