A Tangent on Venus

In last month's blog I chronicled my effort to see Venus as it skirted north of the Sun during its inferior conjunction. It was not a particularly easy task but certainly doable in clear skies. This month I'm doing a little armchair astronomy because that experience made me ponder when circumstances would be at their best to capture Venus as it moves between our home planet and the Sun.

Trigger warning - math is used in this blog post!

While you could spend time scouring the Internet to find the maximum distance that Venus can lie from the Sun during an inferior conjunction, where's the fun in that for anyone with geek tendencies? If we dust off our trigonometry and get some basic orbital elements we should be able to swag an answer to our question. 

It always helps to define some terms before solving the problem. The ecliptic is the plane in which the Earth orbits the Sun. In considering the orbits of other members of our Solar System we can speak of the inclination (tilt) of their orbit relative to Earth's orbit. Those two planes will intersect at two nodes (points). One node (the descending node ☋) would be when the neighboring planet is heading south when it crosses through our orbital plane, while the other (ascending node ☊) it is northbound. What is curious is that there does not seem to be a term for the point 90° farther along the orbit, where the planet would lie maximally above or below our ecliptic. For the sake of discussion, I'll call the point in the orbit where the planet lies at its greatest distance above (north) of the ecliptic as its cresting node and its southern counterpart as its sinking node.

Step 1: How far above the ecliptic does Venus lie when at its cresting (or sinking) node?  

We can readily find Venus' inclination to the ecliptic (3.4°) as well as its average distance from the Sun (a) and then calculate how far above the plane it lies (b) using basic trig:

b = a ÷ cot(θ)

Plugging in to the formula:

a (Venus-Sun distance) : 108 million km

cotangent of 3.4° = 16.8

b = 108/16.8 = 6.4 million km

Step 2: Knowing how high above the ecliptic plane Venus lies at cresting/sinking node and the distance between Earth and Venus at inferior conjunction, we can calculate the angle as seen from Earth for how far above the Sun Venus will appear in the sky: 

θ = tan^-1(b ÷ c)

b (height of Venus above ecliptic): 6.4 million km

c (Earth-Venus at Inferior Conjunction): 42 million km 

tangent ratio: 6.4 ÷ 42 = 0.1524

arctan(0.1524) = 8.67°

Wow - that is almost the length of the upper bowl of Big Dipper. My observation back in January had a distance of 4.8°, a little better than half the best it could be. So having determined the "what", let's turn our attention to "when" we would see one of these inferior conjunctions where Venus is at its cresting node.

This June will mark a decade since the last time Venus transited the Sun. As you may know, these events are quite rare, occurring as a set of transits 8 years apart. In the context of our ecliptic plane and Venus' plane, the inferior conjunction must occur while Venus is moving through its ascending or descending node in order for us to see a transit. And that is why they are confined to a few days around June 8th (as it reaches its descending node) or December 8th (ascending node). If our cresting node can reasonably be assumed to occur midway between June 8 and December 8, the approximate date for it would be on March 8th. As a result, for those of us in the northern hemisphere, an early springtime inferior conjunction will be our best opportunity to catch a glimpse of our sister planet directly above the Sun. 

From a listing of past and upcoming inferior conjunctions, we have the following ones that occur in March:

March Inferior Conjunctions
3/30/2001
3/29/2009
3/25/2017
3/23/2025
3/20/2033
3/18/2041
3/15/2049

Searching online for the inferior conjunction that happens March 23, 2025 we find the difference in Declination between Venus and the Sun will be + 7.5° - adhering pretty well to our predictions. With any luck I'll be around and have a clear day to set up the scope and try yet again to image the event.

A final observation before closing out this post. As discussed in April's 2020 blog when Venus traversed the Pleaides, our sister planet takes just shy of 8 years to return to nearly the same point in our sky. We see that 8 year interval again in the table above where each subsequent inferior conjunction arrives a couple days shy of 8 years. What if we look at the most recent inferior conjunction January 8th and project it out in an unscientific way to estimate its future conjunctions by simply adding 2,920 days repeatedly to it:

January Inferior Conjunctions

1/8/2022

1/6/2030

1/4/2038

1/2/2046

12/31/2053

12/29/2061

12/27/2069

12/25/2077

12/23/2085

12/21/2093

12/20/2101

12/18/2109

12/16/2117

Hmm - the next Transit of Venus is set to occur on December 11, 2117. We didn't land on it exactly because of our estimating shortcut, but there's no doubt that the most recent inferior conjunction is gradually walking the calendar backwards to become the next transit event, gradually losing a little of its northern clearance above the Sun's limb on each conjunction until it finally ends up crossing the orb.

If you have persevered to the end of this geeky blog entry - congratulations! It's probably due in some measure to the fact that, like me and the multitude of ancient astronomers that came before us, you find the patterns and rhythms of the heavens amazing and fascinating to explore.