Four-body problem

A changing pattern made by two or three celestial bodies that move past each other at angles or in curves can be quite complex; what about four?


Such a tango is performed by Venus, Jupiter, Regulus, and the Moon from July 15 to 19. It had to be compressed into one little illustration on page 31 of Astronomical Calendar 2015, but this weblog gives me cyberspace in which to go into it in more detail than you perhaps can bear.

Venus passed Jupiter back on July 1. To launch into detail: the closest approach of the two was at about 4 Universal Time, the conjunction in longitude (when one is exactly north of the other in relation to the ecliptic plane of the solar system) at 8, the conjunction in right ascension (one north of the other as they pass through a telescope’s view) at 14. I’ll abbreviate those three kinds of conjunction as appulse, eclong, and RA.

On July 15 Venus, because its path is curving around southward, has an appulse with Regulus at 6 UT (2.4 degrees southwest of the star), but because Venus is slowing toward the time when it will move backward there is no RA. But there will be an eclong; see the last of the many events of July 19.

By July 16 at 11 UT, the circle into which Venus, Regulus, and Jupiter fit reaches a minimum diameter of 5.6 degrees.

The young Moon, which was down in front of the Sun on July 16, now comes cruising into the picture, first on July 18 passing 4 degrees south of Jupiter (eclong 14:45 UT, appulse 15:27, RA 17:40).


At July 18 16 UT, the circle into which Venus, Jupiter, and the (center of) the Moon fit reaches a minimum diameter of 5.9 degrees.

And at 20 UT the circle into which Jupiter, Regulus, and the Moon fit reaches a minimum of 5.4 degrees.

Then on July 19 the Moon passes only 0.4 degree south of Venus: eclong and appulse virtually simultaneous at 0:51 UT, RA at 1:08. The apparent width of the Moon is about half a degree, exaggerated by 2 in the picture. From northern parts of the Earth it appears a bit farther south. But it occults – hides – Venus as seen from the southern Pacific (diagram in Astronomical Calendar 2015, page 59).

And the Moon passes 3.2 degrees south of Regulus (eclong 0:53 UT, appulse 1:22, RA 2:15).

Not surprisingly it is in the middle of all this, July 19 1 UT, that the circle into which Venus, Regulus, and the Moon fit has shrunk to a minimum diameter of only 3.2 degrees.

Though the curve of Venus’s path never takes it due south of Regulus in RA, it does, just, take it to an eclong (south of the star in relation to the ecliptic), on July 19 about 2:30 UT. And then back in early August. See Venus’s curve (the planet Venus’s curve) in the Venus chart, Astronomical Calendar 2015 page 41.

All this that happens around July 15 to 19 takes place fairly conveniently in the evening sky, about 34 degrees away from the setting Sun.

I learned how to calculate these gatherings in circles from a chapter in Astronomical Algorithms by Jean Meeus, who calls them “trios.” He does not say how to calculate the minimal circle for four bodies; perhaps it is mathematically next to impossible.

You will have noticed another celestial body that might be in the picture but difficult to see.

Now I must get back to what I was doing two or three hours ago when I realized that I had to tackle this, which happened to be thinking out the Venus section for Astronomical Calendar 2016.

5 thoughts on “Four-body problem”

  1. Now that we finally have some clear blue skies with only a few cumulus clouds billowing about, I was finally able to locate Venus today (Saturday, August 8) with my telescope, only 12 degrees from the Sun. It was beautiful to see its razor thin crescent against a blue blue sky with occasional white cotton candy clouds drifting through the scene. Jack has apparently had ample opportunity to view Venus in the daytime from his more northerly location, but I am now finally able to enjoy daytime views as well. A challenge, Mr. Gambino: can you view Venus with your telescope (or binoculars too I suppose) on the day of the inferior conjunction? I have done this on four occasions, but two of them were transits! The two non-transit inferior conjunction passages that I have observed were in October 2010 and January 2006. Good luck sir!

  2. It takes only two to tango, but when you have as many as these, should we call it ballet?

  3. Coming home this evening from the grocery store with my daughter, the western sky revealed a glorious sight, exactly as Guy described above. After arriving home, I was forced to let the frozen goods languish in the trunk of the car while I rushed inside to grab my camera and tripod, with a surprising result once I looked at the downloaded images. Jupiter was nowhere to be seen visually due to the clouds, but appeared as a faint dot in the images . . . Image documentation of Guy’s discussion is available here: (first two images)

  4. Tango, eh? As I’ve sa9d before en francais, La lune et l’ocean, dansent ensemble main en main… the moon and the ocean danse together, Aaaah! Is so beautiful to watch the tide come in as the moon rises.

  5. Just almost more that I could bear but I hung on every word and got more of it than ever before. Great clear and detailed explanation. Thank you very much. I’m SO looking forward to this.

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