Some solutions for homework 3 problems 2-4.

2. Whirlpool simulation 1. (a) At about 210 million years (step 175), a spiral structure begins to develop, becoming clear by 240 million years (step 200). The appearance of the target and intruder galaxies begins to resemble the Whirlpool galaxy and its companion as an apparent bridge of stars connecting the two galaxies develops sometime around 270 million years (step 225). It reaches a "best match" between approximately 300 million years (step 250) and 330 million years (step 300), becomes disrupted by about 420 million years (step 350), and is nearly gone after about 450 million years (step 375). (b) Near the end of the 648 million year time period, the spiral arms begin to dissipate, and many of the stars have apparently escaped from the central nucleus of the target galaxy. (c) In this case, the intruder galaxy passes by in a direction opposite to the orbital motion of the stars in the target. Small, tight spiral arms start to develop around 240 million years (step 200), but they tend to wind back up again and are gone by 330 million years (step 275). The final result is a galaxy with a disrupted disk. No stars escape from the central nucleus of the target galaxy.

3. Simulation 2. Note that the problem parameters used run too fast on most modern machines, and upping the number of stars is helpful. (a) The initial speed of the intruder galaxy nearly matches the orbital speed of the nearby stars in the outer ring. Stars behind the intruder gain energy as they are pulled forward and up out of the plane of the disk. They form the "bridge" of stars that appears to link the target galaxy with the intruder. Stars ahead of the intruder are pulled backward and lose energy and drop into lower orbits with larger angular velocities. As a result, these stars overtake the stars in higher orbits to form a loop of stars. These loops comprise the spiral arm that develops opposite the bridge. (b) In this case, the intruder galaxy passes by in a direction opposite to the orbital motion of the stars in the target galaxy. The intruder spends a relatively brief time near any single star, and manages to pull only a spike of stars from the target. Orbital motion carries this spike away from the intruder galaxy, and differental rotation causes the development and then the winding up a tight spieal features. Less energy is transferred from to the stars in their short encounter with the intruder, and no stars escape from the target nucleus.

4. Ring simulation. Note that the weird "bouncing" phenomenon that sometimes appears complicates execuation. (a) The intruder galaxy pulls rings of stars up and away from the nucleus of the target galaxy. At first these stellar rings contract, and then expand. Such a process, in general, does seem capable of making ring galaxies like the Cartwheel. (b) As the intruder galaxy pulls inward on a star in the ring, the force of gravity does positive work on the star and increases its energy as the star approaches the nucleus. The star's orbit becomes more energetic, with a larger semi-major axis, and it also becomes elongated (non-circular). After reaching perigalacticon (thanks Andy!), the star's new orbit carries it away from the nucleus, and the ring expands. If the star gains enough energy, it may no longer be bound to the target nucleus (you can see this by running the right simulations). (c) A nearly head-on collision is usually required to make a ring galaxy, with in initial x values less than 2. Values of 3 or greater still result in signficant disruption of the target galaxy, and an elliptical ring may be produced by larger values (e.g., x = 7.5). I gave more credit for systematic description of simulations run and the outcomes.