Astr 221 - Homework #7 - due Dec 9

1. Supernovae Light Curves
If the luminosity of a fading supernovae comes from radioactive decay, show that
where lambda is the exponential decay constant of the decaying isotope.

2. The Crab Nebula

The angular size of the Crab Nebula is about 4 arcminutes across, and its distance is approximately 2000 pc. You take a spectrum of the ionized gas in the nebula and find that the Hydrogen Alpha emission line, which is normally at a wavelength of 6563.0 Angstroms has split into two -- one line is at 6594.7 Angstroms and the other is at 6531.3 Angstroms. Given this information, estimate the date of the supernova explosion. Compare this to the observed date of 1054 AD. Why might your number be different (hint: think about physical reasons, not just measurement error)?
3. The Continuing Misadventures of the Starship Asinus

Upon entering the Rigel system, we find not three, but five Hoarsflize warships waiting for us! No planet in sight, either! It's all a trap!!

The Hoarsflize ships begin to fire away at us, led by the enemy battleship Stinger. Luckily (or not), your captain, on the superlight phone with his Antarian bookie who is refusing to pay off the bet on the Super Galactic Bowl, angrily slams down the phone on the navigation console and hits the "warp" button by mistake. Asinus warps randomly to an uncharted part of the galaxy, crippled, but alive. And suddenly "pfutz!" the warp drive has conked out.

But now we are in great need of an Earth-like planet to set down on to repair the ship, and replenish your food and water supplies. We'll also need to find a source for the valuable hydrogen-methane-ammonia mixture which fuels our engines.

There are two different Sun-like stars nearby, but they are in opposite directions, and with the damage to the ship, we only have the range to get to one of them. Our science officer, Commander Ed Yore, finds that both of the stars seem to be moving slighly, a sign that each one has planets around them, pulling on the star and making it move on a very small orbit around the center of mass. In  the language of early 21st century astronomers, the stars are "wobbling."  Unfortunately, your computers were crippled during your last battle with the Hoarsflize, and they cannot analyse the data -- you must figure out which of these star systems to go to. Here is the data on the wobble period and velocity of the two stars due to each planet:

Grayborough's Star
Mewle 809

Planet G1
Planet G2
Planet M1
Planet M2
Period (years)
Velocity (m/s)

(Remember: these are the velocities of the star due to the planets, not the velocities of the planets themeselves!)

Calculate the masses and semi-major axes of each of these planets, and decide which star to go to. Explain your decision.

Now, as you enter the star system, you first go to the Earth-like planet to fix the ship and stock up on supplies (i.e., food -- you are hungry and thirsty). After you have chowed down, it is time to refuel the ship at the other planet. Since you are low on fuel, you must go on the most fuel efficient orbit from one planet to the other -- a Hohmann transfer orbit (remember this from HW#1?). Your navigator, Lieutenant Dawn Key, was injured during the battle with the Hoarsflize, so you must calculate the orbit yourself. Describe the orbit, including the perihelion and aphelion distances, and the orbital eccentricity.   How long would it take you to get from one planet to the other? (Hint: it'll take a while...)

Finally, you are refuelled and ready, after long last, to go back home. But as you prepare to leave the system, Stinger shows up again with guns blazing! Well-armed, Stinger also has shields which absorb all the energy that might otherwise strike the warship.  With your weapons still in need of repair, your only option is to run. Your situation seems hopeless. But then you remember a design flaw in Stinger's shields -- they will blow up once they absorb so much energy that their temperature rises above 2000 K. Not even the Hoarsflize know this (they're not all that smart). You change your orbit to plunge towards the star, with Stinger chasing you (you crafty devil!). How close do you need to get to the star before Stinger blows up?