Radiation Angle of the Project TouCans dipole Part III

 OK, let's maybe wrap this up. Spoiler: I developed more cool new stuff, but didn't wrap this up.

Recap: I set out to see what it would take to map the skips for a QSO, and then try to extrapolate those back to the launch angle of the dipole that KO6BTY and I use for Project TouCans  at our home QTH . Along the way, I learned a lot including another use for the cross product. I also remembered, just now, that dipoles can have more than one launch angle.

Taken from Radio Antenna Engineering

That'll be important later.

Today, as promised, I tried the three skip solution.

Here's the angle of radiation from the SF station with three skips.

But what's the numeric quantity for the angle in degrees? Well, I worked through yet more math. It's getting easier though. I added the code for the cross product of two vectors (position vectors in our case) as well as the usage I'm making of the cosine and sine laws to to earthmid.py.

The numeric result of three skips is 7.714120504450855 degrees.

Here's a quick visual rough check on this

Which, I gotta admit is pretty cool! Yeah, yeah, I know the math should just work, but still!

But was three skips really the answer? Probably not. Why? Check out where the first skip of the four skip solution lands.

A mere fifty miles from the RBN station that picks up Project TouCans VE6JY is the reverse beacon network station with the largest snr value most days. Four skips gives us a launch angle of 14.686256246402552 degrees.

Now, we have an estimate for the launch angle that fits well with our emperical data from SM5CAK and VE6JY.

What else though? Well, have I mentioned K2PO/7 near Portland? The rig hits that RBN station pretty frequently as well. Since it's getting easier and easier to calculate these, let's see where a five skip solution lands

That's not super close, so let's try six skips. I think this might work out better anyway. Really. 

But, nope. I'll give you, it makes for a more cluttered map, but that's not what we were going for.

Let's try 7 skips.... Better!

Let's measure the distance between that landing site and my QTH. It's about 616 miles. The distance to K2PO/7  is about 528 miles. Meanwhile, the distance to the four skip peak projected to the ground is 582 miles. That's the closest so far, so let's go with an eight skip solution.

What's the launch angle for eight skips? Well, take the landing site, and divide that distance by two to find the distance to the first eight skip peak at 41.29982090608962,-120.5723896615752.

We plug that coordinate into our swept angle formula to determine the angle between my QTH and that peak. It gives 3.849189776873525 degrees. We plug that number into the law of cosines method along with the radius of the Earth and the height of the F2 layer in km to get 549.0986270603281 for the length of the third side of the triangle.

We plug that into our law of sines method along with the swept angle and the radius of the Earth to get the third angle of the triangle in this derivation

Note: As I mentioned in the first part, this all going to be quick and dirty back of the envelope stuff until it isn't.

Finally, we plug that angle into the launch angle method along with the QTH to peak swept angle to get a final answer of 34.99157663415465 degrees. And, well, that just doesn't really fit any of the radiation patterns shown above. Except! Did I mention the antennas at a 33 degree angle to the horizontal as it slopes down a hill? OK, there's going to be a part four.

New additions: earthmid.py now has a single method launch_angle_skips that returns the launch angle from the dipole given the receiving station and the number of skips the signal makes.