Calculating the Shortest Distance from an Ionosonde to a QSO Path Using Sage, Cross Products, and Geometry

I've run into an interesting math problem. Project TouCans occasionally makes dx contacts. That's not a problem, that's actually really awesome! However, KO6BTY and I would like to be able to map the F2 skips our radio signal makes on its way to the receiving station. We have plenty of data about the F2 layer of the ionosphere that is captured by ionosondes around the world. The issue is how do we determine which set of ionosonde data to use? Our first guess is that we should use the ionosonde closest to the path of the signal at any point. OK... Now, how do we determine that?

First, we'll calculate the shortest distance from each ionosonde to the path of the QSO between ham radio stations. Then, we can sort the distances so that we can only include the ionosondes that are closest to the path. Armed with that data, we can perhpas used the distances to calculate a weighted, estimated height of the F2 layer at any point along the signals path as it skips from station to station.

Here's an example of the first problem we're trying to solve, namely what's the shortest distance from a given Ionosonde to the path between the two amateur radio stations involved in the QSO?

The yellow line is the path between our station and PG4I in the Netherlands. The red line is the perpendicular line from the ionosonde located in Alpena, Michigan to the path of our signal, (that's the shortest distance between the ionosonde and the signal path.)

But, how do we find the distance along the red line? That problem was handled and explained quite handily here, but I wanted to see the geometry, so we're going to model it using Sage cells on this page. To see the graphical demonstration of each step, click the 'Activate' buttons.

First, let's draw the two station locatiosn and the ionosonde location. Each location is denoted by a vector that extends from the origin (x=0, y=0, z=0) out to the point.

Taking the cross product of the vectors that point from the center of the Earth to the two radio staions defines a plane that contains both stations and the path between them.

The vector result of the cross product is shown a thicker green vector. We're going to use it again in just a moment. You might remember from geometry class that a line perpindicular to a plane defines a family of planes that will all be perpendicular to that original plane. Let's use that to find the perpendicular path from the ionosonde to the path between the radio stations. If we take a cross product of the green cross product vector we just calculated, (remember, it's perpendicular to the plane that contains the two radio stations and the path between them), and the vector to the ionosonde, we'll get back a vector that defines a plane parallel to and the green cross product vector, (and therefore perpendicular to the plane between the stations), and the ionosonde's position vector.

With the above result, you can see that a path from the ionosonde along the plane defined by the cross product in the last step is in fact a perpendicular line from the ionosonde to the path between the QSO stations. The only remaining issue is that even though we can see the point where the intersection occurs, we don't know where it is. We don't have a numeric latitude and longitude. One more cross product will provide the answer. If we take the cross product of the two vectors produced in the last two steps, we'll get a vector that has to be perpendicular to both of the original vectors. Since those vectors are perpendicular to the two planes they defined, the only place the result of the new cross product can exist is along the intersection of the two perpendicular planes! Here's what that looks like:

I've taken the liberty of converting the Cartesian coordinates back to Earth based spherical coordinates. From that point, it's a simple matter to use the haversine plugin in Datasette to determine the distance from the Ionosonde to the path of the QSO. If you're curious about the code that created these demonstrations, I've stashed it away at the repo for the ham radio and Reverse Beacon Netowrk logger KO6BTY and I have been working on.

The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1. While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero. This is typically explained in the following way. While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare. That doesn't stop us from looking for them though! Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2]. Cabrera was utilizing a loop type of magnetic monopole detector. Its operation is in...

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

Now available as a Kindle ebook for 99 cents ! Get a spiffy ebook, and fund more physics The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems , there seem to be fewer explanations of the procedure for deriving the conversion, so here goes! What do we actually want? To convert the Cartesian nabla to the nabla for another coordinate system, say… cylindrical coordinates. What we’ll need: 1. The Cartesian Nabla: 2. A set of equations relating the Cartesian coordinates to cylindrical coordinates: 3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system: How to do it: Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables. The chain ...

More Cowbell! Record Production using Google Forms and Charts

First, the what : This article shows how to embed a new Google Form into any web page. To demonstrate ths, a chart and form that allow blog readers to control the recording levels of each instrument in Blue Oyster Cult's "(Don't Fear) The Reaper" is used. HTML code from the Google version of the form included on this page is shown and the parts that need to be modified are highlighted. Next, the why : Google recently released an e-mail form feature that allows users of Google Documents to create an e-mail a form that automatically places each user's input into an associated spreadsheet. As it turns out, with a little bit of work, the forms that are created by Google Docs can be embedded into any web page. Now, The Goods: Click on the instrument you want turned up, click the submit button and then refresh the page. Through the magic of Google Forms as soon as you click on submit and refresh this web page, the data chart will update immediately. Turn up the:...

Copasetic Flow

Search This Blog