The Coral Model of Colonization. We can estimate the time itwould take for a

Chapter 13, Problem 56

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The Coral Model of Colonization. We can estimate the time itwould take for a civilization to colonize the galaxy. Imaginethat a civilization sends colonists to stars that are an averagedistance D away and sends them in spacecraft that travel atspeed v. The time required for travel, is thenSuppose that the colonists build up their colony for a timeat which point they send out their own set of colonists to otherstar systems (with the same average distance and same spacecraftspeed). Then the speed at which the civilization expandsoutward from the home star, (for the speed of colonization),is However, this is true only if the colonizationis always directed straight outward from the homestar. In reality, the colonists will sometimes go to uncolonizedstar systems in other directions, so we will introduce aconstant k that accounts for this zigzag motion. Our equationfor the speed at which the civilization expands outward fromthe home star isFor the purposes of this problem, assume that and thatthe average distance between star systems is light-years.a. How fast (as a fraction of the speed of light) does the civilizationexpand if its spacecraft travel at 0.1c and each colonybuilds itself up for 150 years before sending out the nextwave of colonists? How long would it take the colonists toexpand a distance of 100,000 light-years from their home starat this rate?b. Repeat part (a), but assume that the spacecraft travel at 0.01cand that each colony builds itself up for 1000 years beforesending out more colonists.c. Repeat part (a), but assume that the spacecraft travel at 0.25cand that each colony builds itself up for 50 years before sending