What happened after the Small Bang?

Question

In Jan 1st 2000 infinitely many space ships were fired from Earth, all in the same plane (that of MIlky Way). Afterwards all ships continued to move with constant velocities. Initial velocities of space ships were randomly and uniformly distributed, the meaning of 'randomly and uniformly' is explained in the next paragraph.

After 10 years in journey by their own clocks each captain of each ship observed that there are approximately 1000 other ships, give or take, uniformly distributed within distance 0.01 light year of their own ships.

The base on Earth also drew a map of current position of each ship on Jan 1st 2010.

http://i36.tinypic.com/2vtxif7.jpg

Obviously all ships appear within a circle ABC of radius

R = 10 light years.

How many ships are within triangle ABC?

Answer 1

Answer -

Total number of ships X area of triangle / area of circle

Answer 2

How many fish are in the ocean if you draw a triangle?

http://home.comcast.net/~eschermc/Circle...

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The angles all go to zero on the hyperbolic plane.  Perhaps I can stand on the shoulder of the great mathmatician Duke on how to solve this interesting problem.

................Addendum 1..................

There will be a lot of ships in ABC.

This is complicated. With an infinite number of ships, limited only by the speed of light, you are going to get a distribution quite similar to the fish in the Escher drawing I cite above.   This will skew the number of ships in a given area.  There will be far more than 1000 per π*0.01^2 ly^2 as you approach 10 ly from earth (as measured from earth).  (Regarding the distribution see my answer to a prior problem:  http://answers.yahoo.com/question/index;... )

The way to solve this is to look at it from the perspective of the ships captains.  All captains will see ships both in front and behind them.  Further, when the captains are at 10 yrs their time, each of them will report a distribution identical to what is seen by earth at 10 yrs earth time.  There is no center.

Moreover, some of those ships (actually an infinite number) will be, by earth's clock, nearly an infinite distance and infinite time away when those ships look back and say that they are 10 ly from earth.  Special relativity length and time paradox.

But, you can take a snapshot of when each ship -- by its own standards -- reaches 10 ly away, and use that to compute the density function.

If you do this, you will get a set of circles that extend to infinite in size.  But all the points inside will have the right density.  You will also note that the "topographical" lines connecting points A, B and C will be curved and that A, B and C will have zero angle.

It looks to me like you are dealing with hyperbolic cruves.   (Perhaps Zo Maar will be able to solve this).  To solve this assume that ABC is a hyperbolic triangle going to infinity at A, B and C.   But it will constitute a fixed portion of the overall area of the 10 ly circle from earth.  Moreover, there will be a "defect" based on the psuedo curves.

Now, go to this website:  http://www.geom.uiuc.edu/java/triangle-a... .  Use the applet at the bottom to figure the area of the triangle in relationship to the overall area of the hyperbolic circle at infinity.  Just move 3 vertices to the edge of the circle.  Pretty cool.

It turns out that the answer for our area is:

= k^2[π-(α+β+γ)]

Here, α,β and γ are the respective angles for A, B and C.  Moreover, they go to zero and infinity therefore the answer is

= k^2 *π

Which is the maximum area of a triangle in hyperbolic

I will assume that k= 10 ly, therefore the area would be

3.14 x 10^4 ly^2

Given a density of 1000 per π*0.01^2 ly^2, the answer is

10^11 ships.

Very challenging -- I learned a lot.

Answer 3

You say "the meaning of 'randomly and uniformly' is explained in the next paragraph."  but it isn't.

Answer 4

The concentration or density of ships is 1000 per π*0.01^2 square light year.

So for a total area of π*10^2, there are 1000e6 ships, or 1 billion ships.

Since the distribution is uniform in space, the number of ships in that triangle is the fraction of the area of the triangle to the circle multiplied by 1 billion.

The dimensions of the triangle are not given, so I doubt you can go any further without guessing something.

Answer 5

Probably all of them, unless line AB is very close to earth.