Can the period of the planet transit across the sun be calculated in the same way the eclipse year is?

Question

I read from Wikipedia$-$Eclipse cycle that the eclipse year, which is the period of Earth's hitting a certain node of its orbit around the sun, that is, the ecliptic, and the moon's orbit around Earth, can be calculated as the beat period of the synodic month and draconic month. The recent occurrence of planet transits across the sun lets me wonder if this principle can be applied to the case of a planet's transit across the sun as seen by an observer on another planet in an outer orbit.

When Venus transited across the sun on June 6, 2012, I read it takes place in pattern of the 8-year-interval pairs with each pair at the interval of 105.5 years and 121.5 years alternately, with one in each pair being when Venus crosses the ecliptic plane at its ascending node and the other being when Venus crosses the ecliptic plane at its descending node. This made me contemplate whether the period of the Venus transit across the sun when it crosses the ecliptic plane at its certain node can be calculated in the way the eclipse year is, that is, whether the following holds:

The period of the Venus transit across the sun as it crosses the ecliptic plane at its certain node = The beat period of the period of Venus' being in inferior conjunction with the sun and the period of Venus' hitting that node.

Early this July I read from EarthSky an observer on Pluto may, through some optical aid, be able to see Earth transiting the sun on July 12, when Pluto was at opposition, which, together with the upcoming event, that on October 24, 2018 Pluto will cross the ecliptic plane at its descending node, is responsible for the Earth transit as seen from Pluto. Last time this kind of transit occurred on January 11, 1931 on account of both that Pluto was at opposition that day and that Pluto crossed the ecliptic plane at its ascending node on September 9, 1930. And next time this kind of transit will occur when Pluto is at opposition 161 years later, around when Pluto will cross the ecliptic plane at the same node it crossed in 1930. This makes me again contemplate whether what follows holds:

The period of the Earth transit across the sun as seen from Pluto as Pluto crosses the ecliptic plane at its certain node = The beat period of the period of Pluto's being at opposition and the period of Earth's hitting that node.

I think the principal difference between the eclipse year and planet transit across the sun is that in the former case both the moon and the sun are in a closed orbit around Earth while in the latter case only the sun is in a closed orbit around the observer-sited planet (the transiting planet is in a tortuous path from the perspective of the observer). I don't know if this feature would affect the applicability of the beat-period principle. If the principle of beat period as it used to calculate the eclipse year can be applied to the above two examples, I think it should be able to be generally applied to calculate the period of any planet's transit across the sun as seen by an observer on an outer planet, right? That is, should what follows generally hold?

The period of a planet's transit across the sun as seen from an outer planet as it crosses the outer planet's orbit plane at its certain node = The beat period of the period of the transiting planet's being in inferior conjunction with the sun from the outer planet's perspective and the period of the transiting planet’s hitting that node

Answer 1

I apologize for the length of this in advance! Before getting to the answer it should be said from an orbital mechanics point of view all transits are eclipses and all eclipses are transits. There are a lot of different types of both and the words are used differently but they mean the same thing - something is passing in front of something else from the observers point of view. Not many would say "Venus eclipsed the Sun" during a Venus transit nor would they say, "the Moon transited the Sun" during a solar eclipse. But the way cycles are determined is the same for eclipses or transits.

In short, the answer to this is almost yes:

The period of a planet's transit across the sun as seen from an outer planet as it crosses the outer planet's orbit plane at its certain node = The beat period of the period of the transiting planet's being in inferior conjunction with the sun from the outer planet's perspective and the period of the transiting planet’s hitting that node

For us to see Venus or Mercury transit/eclipse the Sun, three things have to happen. The inner planet's nodes must be at the same ecliptic longitude as the Sun, the planet must be crossing the node when this occurs, and it needs to be on the same side of the Sun as us. (Oh, and the observer needs to be on the Sun side of the Earth with clear skies which is NOT what happened to this observer during the last Venus transit and I'm still bitter about it.)

Various cycles of different lengths have to come together to make this happen which is what I think you mean by "beat period" though I've never heard it phrased this way in astronomy. It's as if I ate dinner on Friday every two weeks with one friend and every three weeks with another - then the "beat period" of six weeks would mean a scheduling conflict or a dinner party.

The reason I said your conclusion was almost correct is explained by looking at the transits of Mercury on this Wiki page. Sometimes a transit doesn't happen even though it is on the "beat period" (which astronomers just call cycles). Lots of smaller cycles line up, or come together, every so often on a larger cycle. You got it right when you said the two things have to happen - the nodes lining up with the Sun during an inferior conjunction but as Mercury shows the transit can be missed. If a larger "beat period" was used to leave out those missed transits, there would be extra transits happening mot on the beat if that makes any sense.

Determining these cycles is fairly easy for us on Earth due to the frame of reference of the ecliptic. Neither the Earth nor the Sun have "nodes" to the plane of the ecliptic because it's the Earth and the Sun which define the ecliptic. Nodes are where tilted (inclined) planets cross this plane. It's a handy reference since we live on Earth, but for the observer freezing on Pluto wondering about all of the transits they will get to see with their telescope, they would want the ecliptic be the plane defined by Pluto and the Sun. Then the Earth and all of the planets would have nodes as they crossed above and below the plane of the Pluto - Sun ecliptic. From their perspective, there would be inner planet cycles which would repeat with a "beat" when certain cycles lined up again so that Jupiter, for example, transited the Sun.

On the Wiki page you linked talking about eclipses, I found it awkward the way they kept referring to the "Sun being near/at the node" which technically isn't wrong because we know they mean "from the observes viewpoint, the planet's node lines up with the Sun (same ecliptic longitude)" but I think it could give people the wrong impression as if the Sun was at or near that location in 3D space. That's just my opinion, though.

Here is a good Sky & Telescope article on Venus transits and how they will become more rare in the future. And here is one of the most enjoyable websites on the internet - Orbit Simulator. Look through the list until you see something that interests you and then click it to see fantastic browser based orbit details. They also have a Windows program you can download to have a lot more control over the simulations.

I apologize again for the lengthy reply - orbital mechanics is a subject I love. And good on you for putting so much thought into the orbits and cycles, not many want to dive that deep into the subject.