A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2019 May 16, 16:31 -0700
There's an easy way to solve this problem that is also easy to visualize. You're asking for the date and time when a moving celestial body, Saturn, is aligned along the same great circle with two fixed stars. Let's bring this down to Earth...
Suppose I am sailing along in my schooner, and I see two lighthouses off starboard. One lighthouse is two miles away. The other is eight miles away. The nearer one is to the left of the more distant at first, but the gap is closing. As I sail along, I realize that a moment will come when the two lighthouses are aligned. One will disappear behind the other, and at that instant, my schooner and both lighthouses will be aligned along one straight line. All I have to do to predict this time is calculate the bearing angle (azimuth) of each lighthouse as seen from my vessel at regular intervals of time. At some point in time, the lighthouse on the left will become the lighthouse on the right, and I can interpolate in between to get the exact moment when they are aligned.
Now scale it up. I'm sailing along in the mid-Atlantic, and I'm curious to know when Bermuda, which is 200 miles to my WNW will be exactly aligned with Cape Hatteras which is about 800 miles away. It''s the same problem, except that now I may want to do the math by great circle bearing (not really necessary at these relatively short distances but it can't hurt). So you calculate the bearing (true azimuth) of Bermuda from your vessel, and you calculate the bearing of Cape Hatteras. Interpolate, as above. When the initial great circle bearings match, the vessel and both fixed locations are aligned along the same great circle
On the celestial sphere, it's exactly the same problem with Deneb and Altair replacing the fixed lighthouses and Saturn replacing my schooner. To switch things up a bit, let's calculate the great circle bearing of Altair as seen from Deneb. This is called a "position angle" in astronomy, but that's not important here. It's just the bearing across the celestial sphere of Altair relative to Deneb. Then calculate the bearing of Saturn seen from Deneb at regular intervals. When they match, all three bodies are on the same great circle.
Yes, it's that easy. Just dig out your favorite method for calculating initial great circle bearing. Work that for Deneb to Altair. Then work it for Deneb to Saturn at midnight UT each day (or some convenient daily time). At some point the difference in bearing between Altair and Saturn as seen "from" Deneb will flip sign. Interpolate to get the exact time when the difference in bearing was zero. It's just as if we are observing from lighthouse "Deneb" and Saturn is a travelling ship off in the distance beyond that "lighthouse" Altair.
Note that this approach does not care if the stars are fixed on the celestial sphere. This allows you to use their slightly shifting relative positions (due to aberration). And for extra credit, does that make even one second difference in the time of exact alignment? Two seconds??