Mystery "missile launch" off California – solved?

Not a missile launch.

…

Here’s the idea with math:

The idea that it’s a missile launch comes from three misconceptions. Firstly that the trail is vertical – it’s not, it’s a horizontal trail, at around 32,000 feet (about six miles). It’s the same as this:

This contrail is no more vertical than the road is, and nor are the power lines at 45 degrees. Everything is horizontal – it’s the just the angle you are viewing it from. All of these show horizontal contrails.

Secondly there’s the misconception of direction, that it’s flying away from the viewer, when it’s actually flying towards the viewer. This is because the “base” of the contrail seems wider than the tip. Perspective tells the brain that this mean the base is closer. But actually you can see the base has been greatly spread by the wind. Since it’s still so far away the effects of perspective are greatly diminished, meaning the actual width of the contrail is what is creating the illusion. Imagine is a plane with a 100 mile long spreading contrail were coming towards you; what would it look like? It would look exactly like this.

Thirdly there’s the idea that it goes all the way down to the ground. Now that might be true if the Earth was flat, but the Earth is round, and things go beneath the horizon eventually, no matter how high they are. A plane 200 miles away but five miles up is always below the horizon. If the horizon is raised (as it is here, with Catalina Island), then the distance is less. Here’s some math:

This diagram is not to scale, but the math is the same regardless. The solid curved line is the surface of the earth. The dot at the top is San Clemente. The little triangle is Catalina. “d” is the distance to Catalina (d=35 miles). “c” the amount of Catalina that is visible above the horizon (c=0.05 miles, really a bit more, but let’s be conservative). “a” is the altitude of the plane, (a = 6 miles). “r” is the radius of the earth (r=3963 miles).

The green wavy line is the contrail. Notice it’s at a fixed height above the surface of the earth, and is going directly towards the OC.

The point labeled (0,0) is the center of the earth. (0,0) means X=0, Y=0, where X is horizontal and Y is vertical. What we want to know is how far away the plane is, the value x. We do this with cartesian geometry, noting that the lowest visible point of the trail is at the intersection of the dotted line, which is a circle of radius (r+a), hence the equation x^2 + y^2 = (r+a)^2 and the line labeled “sight line”, which is has the equation y=x*c/d. Combining these equations to solve for x yields a quadratic equation, which we can solve with Wolfram Alpha:

intersection of (y=r+x*c/d) and (x^2+y^2 = (r+a)^2)

and with the real numbers:

intersection of (y=r+x*c/d) and (x^2+y^2 = (r+a)^2) where a=6 and d=35 and c=0.05 and r=3963

Which gives x = 212, meaning that the bottom of the contrail is around 200 miles away. So if the front of the contrail (the actual aircraft) is somewhere above and behind catalina, then that means the contrail is over 100 miles long. At 500 mph, that means it could have formed in 12-15 minutes, which seems consistent with the descriptions in the discussion above. (feel free to play around with the numbers there to see the affect of various assumptions)

Full post here: Jet contrails from some angles look like missile trails