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.

Diary of a TOV Chaser

Jun 4th - 4 p.m.: A few months before the 2012 transit of Venus I promised myself that I'd make every effort to catch this rare alignment. I begin checking various weather sites and Clear Sky Clock pages, and prospects for viewing the transit appear iffy at best for the mid-Atlantic region. North and south don't appear encouraging either, but Kentucky and westward looks like a winning strategy. It seems as though a road trip is in order!

Jun 5th - 5:00 a.m.: Plans are to make it somewhere between Lexington KY and Evansville IN, roughly an eight to ten hour drive. Scope and my newest asset, an ImagingSource video camera, are packed in high hopes of getting some footage of the historic transit.

2:00 p.m.: Pulling into Lexington for lunch the skies look pretty reasonable, lots of puffy cumulus scattered about. Using my cell I check live camera shots and CSC pages for Louisville and Evansville (a distinct advantage over the intrepid ToV astronomers of old trying to navigate to a clear venue). With only incrementally better skies farther west within a few hours' drive I elect to throw in my lot with the local astronomy club that is holding a transit viewing at the University of Kentucky's arboretum.

3:00 p.m.: As I pull up to the arboretum I immediately spy a couple of scopes set up on the front lawn. I park and go over to introduce myself to the other amateur with the scopes, assuming he's the advance guard for the club's ToV event. Turns out Steve is actually from Newport News in Virginia, and like me had headed west in search of blue skies. While I set up the scope we chat, trading stories of our various astronomical experiences and our hopes to add yet another that afternoon.

4:00 p.m.: WKYT, the local TV station, shows up to get a 'teaser' about the event at the arboretum. We protest that neither of us can officially speak for the club - but fortunately most questions involve the transit which we are able to answer. Two hours to go - are we over-eager or are the club members very relaxed about getting in place for the viewing?

4:30 p.m.: The person in charge at the arboretum pays us a visit. "We are getting a lot of calls about this, it's generating a lot of interest. Be careful where the rest of your group sets up because the lawn serves as overflow parking." Our group? We nod and assure her it'll be fine. Looks like the club has done a solid job getting the word out for the public to come witness the transit.

5:00 p.m.: The first few club members arrive, along with a gradual influx of Lexingtonians. We ask them who is bringing scopes since they did not. The answer is vague. The sky is also beginning to slip with the cumulus giving way to a more ominous altostratus blanket. Noooooo!

5:30 p.m.: Now I'm worried - we have three scopes on the field (Steve's two and mine) and an ever growing inquisitive crowd. I begin to strategize that if I can manage to carve out the ingress event for my own video efforts then afterwards I'll jump in and do outreach. I get the laptop on and begin to try to get the equipment ready. The Sun spends increasingly longer times obscured by the clouds, but hope springs eternal.

5:50 p.m.: WKYT is back with a different crew and they want more footage for their news at 11. My ToV tee-shirt seems to strike a chord with them, along with the fact that Steve and I have traveled hundreds of miles to get here. "Why would you do that?" is the first question I field as the camera rolls. I do my best to channel Carl Sagan, but the whole time I'm thinking "get this over, I have a transit starting momentarily and it won't wait!"

6:03 p.m.: The interview is over but the transit is eminent as the Sun pokes through a break in the clouds. The laptop has decided to hibernate while I was interviewed and the rough polar alignment has lost the Sun. Firing up the laptop serves as a magnet, drawing the crowd - which by now is probably 100+ people - to watch over my shoulder as I struggle to recover. First contact has passed and I am still trying to get things in place when I suddenly get it. I'm here for a reason. I had thought it was to come back with artifacts of a once in a lifetime event, but actually it's to relax and share it with scores of people whose curiosity has brought them out to this field. I grab my 7x50 binoculars, ensure that the filters are secure, and grab a quick look at a fantastic image - Venus has just cleared the limb at about the 1 o'clock position, a tiny paper punch hole in the solar disk.

7:45 p.m.: I'm hoarse and my lips slightly chapped from all the talking, fielding questions from how the telescope works to whether we'll be back tomorrow night. The crowd has thinned as we have not seen the Sun since about an hour ago. But we had a good run, and in those 40 minutes I helped everyone from grandmothers to preschoolers get a chance to see this rare event. The binoculars worked incredibly well as people had an instinctive sense of how to use them, and the Orion filters provided a naturally orange look to the Sun. Steve had also abandoned most of his efforts and turned to showing people the view through his H-alpha scope and projection device which he fashioned from an S&T article. I bid safe journey to my new friend and head back towards Baltimore.

 

11:20 p.m.: From my hotel room in eastern Kentucky I stay up to catch the 11 o'clock news on WKYT. Sure enough, there's a segment on the event at the arboretum with the crazy folks who traveled hundreds of miles for a glimpse of Venus crossing the Sun. And there I am, trying to explain it, my 15 seconds of fame. I smile as I turn off the set and climb into bed. I wanted a memorable ToV and I sure got it. As it is written, "Man proposes, but God disposes"!