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Below this answer to Why are Delta Aquariids “for the southern hemisphere” while the Perseids are “for the north”? I wrote the comment:

+1 To make this complete, Comet Swift-Tuttle has a nearly circular orbit of about 1 AU but an inclination of 113.45° which is just about what it would have to be (double) to push the Perseids' radiant to a declination of +58°.

Now I'm thinking that a vector that points toward's a meteor shower's radiant is just the vector sum of the Earth's orbit's velocity vector and the velocity vector associated with the orbit of the shower's associated comet at their intersection point where the shower is a maximum.

Question: If that's the case, is there a simple set of equations for the position of a meteor shower's radiant point and a source for them that can be cited?

It seems like this would have been done quite a long time ago and should be found in some classic early work!

note: As discussed in comments, rewriting "just the vector sum of the Earth's orbit's velocity vector and the velocity vector associated with the orbit of the shower's associated comet at their intersection point" as $\mathbf{\vec{r}} = \mathbf{\vec{e}} + \mathbf{\vec{c}}$ doesn't help as we still need the same amount of information.

I wonder if the early work on radiant celestial coordinates and comets contain some interesting math based on Keplerian elements - we have the information that the two orbits intersect and we know where it happens (because of the date) so there is substantial reduction in the number of independent variables.

For additional reading and resources about the relationship between a meteor shower and it's associated comet's orbit, see:

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  • $\begingroup$ I'm not clear on what you're asking for, as it's hard to get any more simple than $ radiant = \vec{e} + \vec{m} $. If you're looking for sources, you might have better luck searching for information on computing the parent body's orbit from the radiant, because that's the more common problem. We observe the meteor direction and velocity, subtract off the Earth's velocity, and compute the parent body's orbit from that. $\endgroup$
    – Greg Miller
    Commented Jul 2, 2023 at 4:42
  • $\begingroup$ @GregMiller hang on I'll take a look right now (it's been 3 years...) $\endgroup$
    – uhoh
    Commented Jul 2, 2023 at 4:42
  • $\begingroup$ @GregMiller I did write "Now I'm thinking that a vector that points toward's a meteor shower's radiant is just the vector sum of the Earth's orbit's velocity vector and the velocity vector associated with the orbit of the shower's associated comet at their intersection point where the shower is a maximum." and later "Question: If that's the case, is there a simple set of equations for the position of a meteor shower's radiant point and a source for them that can be cited?" $\endgroup$
    – uhoh
    Commented Jul 2, 2023 at 4:45
  • $\begingroup$ @GregMiller So it looks like the title doesn't do a good job of summarizing the actual question that I've asked (and readers should read!). I'll update the title to better match the actual question post. Is there anything about the post itself that leaves the question still unclear for you? I'll add the post that r = e + c (c=original comet) isn't really a "simple equation" to me in this context since there is still the problem of finding e and c. Give me an hour; I just asked a long question elsewhere and need some oxygen + coffee. $\endgroup$
    – uhoh
    Commented Jul 2, 2023 at 4:46
  • $\begingroup$ While finding "e" and "c" are far from trivial, they are solved problems. For example, the velocity of the Earth can be obtained using the JPL Development Ephemeris, or VSOP87. The position and velocity of the comet partcles can be obtained by solving Kepler's equation. I'm not sure it's worthwhile to repeat how to do those here. $\endgroup$
    – Greg Miller
    Commented Jul 2, 2023 at 15:02

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