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The Backtrack Orbit Search Algorithm: The Pittsburgh Problem

The General Case

The Pittsburgh Problem is, more descriptively, the question of what happens in the Backtrack Orbit Search Algorithm when the point in question is right at the inflection point of an orbit.

What happens at the inflection point is a special case. I'm going to go through the more ordinary case first. The picture below is a relatively ordinary orbit for a polar orbiter. The ascending equatorial crossing is at 77.75E. The satellite has a declination of 8.78 degrees and a period of 101 minutes. The swath width is 1400 km.

We want to find all the orbits for which this sensor sees Pittsburgh on the ascending pass. Given the location of Pittsburgh (40.50, -80.22) the algorithm looks at what the swath would look like if the groundtrack of the satellite went right through Pittsburgh on the ascending pass. For the example I'm just eyballing it - this orbit has an ascending crossing of about -70.1.

The algorithm takes that swath and figures the longitudinal half swath widths in either direction from Pittsburgh. In this case those are about 8.56 degrees West and 8.19 degrees East. That means an orbit whose groundtrack passes through the point (40.5, -71.66) will see Pittsburgh on the western edge of the swath during the ascending pass. That orbit has an ascending equatorial crossing of about -61.4

And an orbit whose groundtrack passes through the point (40.5, -88.41) will see Pittsburgh on the eastern edge of the swath on the ascending pass. That orbit has an ascending equatorial crossing of about -78.2.

So for any orbit with an ascending crossing in the range [-78.2, -61.4] the sensor will see Pittsburgh on the ascending pass.

The Special Case

What Ted has labeled "The Pittsburgh Problem" is just a special case. Ted wants to know what happens if Pittsburgh is on the inflection latitude of the satellite. I've create just such a satellite and a rather typical orbit from that satellite appears below. The sensor still has a swath width of 1400 km. and the satellite still has a period of 101 minutes. But the declination is now 49.5 degrees so the inflection latitude is 40.5 degrees.

Again, given the location of Pittsburgh (40.50, -80.22) the algorithm looks at what the swath would look like if the groundtrack of the satellite went right through Pittsburgh on the ascending pass. This orbit has an ascending crossing at about 15.3 degrees.

Again the algorithm takes the swath and figures the longitudinal half swath widths on either side of Pittsburgh. To the East that's about 40.47 degrees. To the West ... its zero. We are at the inflection point. Beyond this point the orbit is descending, but we're looking for orbits that see Pittsburgh on the ascending pass. This is the orbit that sees Pittsburgh on the western "edge" of the ascending pass.

For the minimum crossing we look at the orbit whose groundtrack passes through the point (40.5, -120.69). That orbit sees Pittsburgh on the eastern edge of the ascending pass and has an ascending crossing of about -24.8 degrees.

So for any orbit with an ascending crossing in the range [-24.8, 15.3] the sensor on this satellite will see Pittsburgh on the ascending pass.