----- Original Message ----- From: "Erik Reuter" <[EMAIL PROTECTED]> To: "Brin-L" <[EMAIL PROTECTED]> Sent: Monday, January 07, 2002 9:21 PM Subject: Re: another asteroid near miss
> On Mon, Jan 07, 2002 at 07:51:06AM -0600, Dan Minette wrote: > > I did a small back of the envelope calculation on this. There should be, > > roughly, 1 asteroid that hits the earth for every 10,000 that passes within > > 400,000 miles. > > > This seemed like a fun calculation, so I tried it, assuming a 4000 mile > earth radius. I got 20,000:1 odds. Did I make it an error? > > For every asteroid that passes within R miles of the earth, consider > the instant when the asteroid first intersects the sphere, centered at > the earth, with radius R. Assume the radius of the earth is r. Then the > earth subtends a solid angle of Pi*r^2/R^2 for the asteroid at that > instant. The asteroid is equally likely to be heading in any solid angle > of 2*Pi steradians (since it is going into the sphere, not out, it is > half of 4*Pi). This is where I think you make an error. You assume that the direction of asteroids who's closest point of approach to the earth is <=R can be represented at a point in time by an half isotropic source of asteroids at a distance R. > > Therefore, the probability P that the asteroid hits the earth is > > 2*Pi / (Pi*r^2/R^2) > > 1/P = 2 ( R / r ) ^2 or 20,000:1 (actually, 19,999:1 if we are picky). > > Is there an error in my thinking? > Let me use that same calculation twice to show that, together with the assumption made above, it comprises a system with internal inconsistency. Let us assume that R is 40,000,000 miles. Let us assume that there is indeed an isotropic source of asteroids in a thin spherical belt at that distance. Let us also neglect the effects of gravity (at 40 million miles this is certainly invalid, but it does help illustrate the difficulty in the technique). Let us look at two cases, where r1=400,000 miles, and where r2=4,000 miles. For r1, we have 1/P =2*100^2=20,000, for r2, we have 1/P=2*10000^2=200,000,000. The ratio of these two numbers is 10,000, not 20,000. Since you did the math OK, I think the problem is assuming a half isotropic source. The way I looked at the problem is noting that the, at the position of closest approach to the earth, the velocity of the asteroid is orthogonal to a line drawn from the earth to that asteroid. I looked at a plane that contains that line and is perpendicular to the velocity of the asteroid. Its an interesting problem, and not as simple as it might appear at first glance. Dan M.
