Locus

I’ve found a clear way of showing how Jupiter swells brighter and slightly less bright, stands higher and lower, over its twelve-year cycle. At least I hope it’s clear.

If you were looking at the Moon through your moon window around two nights ago when it was Full (the night between Feb. 3 and 4), you will have seen a star above it and a star a handspan to the left. The star above was Jupiter and the star to the left was Regulus. The Moon, now beginning to be “decrescent” – that is, with a subtle softening by shadow around its right side – has moved on, passing south of Regulus last night.

Now comes the opposition of Jupiter – the great event of this part of this year, the opposition of the greatest body that can have an opposition. Or we could say that the Anti-Sun position in the sky swept over the Moon where it was two nights ago and now sweeps over Jupiter.

The opposition happens in this coming night between Feb. 5 and 6, at 6 hours Universal Time, which is 6 AM in Britain, 1 AM in the eastern US, and on the west coast 10 PM in Feb. 5. The exact moment matters only theoretically; it’s merely the middle of the good time for observing the planet, and that good time extends two months or so on either side.

You could say that the good span for seeing a planet is roughly the time when it is retrograding (appearing to move backward, because we are overtaking it) minus the times of Moon-glare within that span. Jupiter is face-on to us now but outcompeted by the Moon; observing it will get easier over the next week or two as the Moon gets farther east and rises later.

There is a lot more in Astronomical Calendar 2015 about Jupiter and its opposition, but let’s add something.

When, only six days ago, the subject was Juno (the asteroid named for Jupiter’s wife), I showed you an improved version of my small chart of the “locus” of Juno’s oppositions. Locus is Latin for “place,” but it’s used in mathematics to mean a line connecting all points that meet a certain condition. I was first interested in this when I learned from Jean Meeus’s book that the locus of the places in the sky where Juno can be at opposition happens to form an almost straight line along the sky’s equator. I had already realized that the meaning of this “locus” could be made clearer by including not only the locus curve but the asteroid’s track for a span of time around some actual opposition dates, showing that the midpoint of this track is always on the locus curve.

The same kind of chart can show that other asteroids have opposition-locus curves very different from Juno’s; and can be used not only for asteroids but for the planets. Since the planets orbit in planes close to the ecliptic plane, their opposition-locus curves will depart not much north or south from the ecliptic, and their opposition tracks will just about run along in their locus curves.

However, I realized that the chart for a planet, or any body, could be made richer by including not only the locus curve and the good-observable-time tracks, but symbols (circles) for the body at the opposition dates. These symbols can be graded in size, like those for stars – but, finally, sized not just for brightness but for difference in brightness from the body’s average opposition brightness; or, better, brightness compared with the range between its brightest and dimmest oppositions.

Thus results a useful chart in which, for Jupiter’s oppositions over its twelve-year cycle, you not only can see where they happen but can simultaneously compare their declinations and their peaks of brightness.

JupOppsRt

JupOppsLt

Declination makes the greater difference, because it is easier for us northerners to watch Jupiter when it is 22.7 degrees north as in 2014, than when it is 22.4 degrees south as in 2019. But it’s interesting to see that the peak of brightness is magnitude -2.9 when near Jupiter’s 2011 perihelion and -2.5 when near its 2017 aphelion. We could as well give a figure for Jupiter’s angular size in the telescope, which, like brightness, depends on its distance.

The locus of the oppositions is drawn in yellow, and you can see where it curves slightly north and south of the ecliptic. The tracks, drawn in black, are for the 200 days centered on opposition, so as to include the 121 days when Jupiter is retrograding (e.g. from its 2014 Dec. 8 turn back westward to its 2015 Apr. 8 resumption of forward motion). The little arrowhead at the end of the track shows how far Jupiter has retraced its steps forward by the end of the 200 days.

The boundaries of the zodiacal constellations are shown, without their names, which would be too cluttering. Perhaps you can recognize them. Jupiter at its present opposition is retrograding from Leo into Cancer; in 2016, it will be well over in Leo; 2017, in Virgo; 2018, in Libra; 2019, in non-zodiacal Ophiuchus.

For asteroids and comets that don’t hug the ecliptic, these retrograde loops will look clearer! So we’ll get back to locus charts sometime.

10 thoughts on “Locus”

  1. This is interesting, thank you. I’m curious how you determined the relative sizes of the circles for Jupiter’s brightnesses at the various oppositions. To my eye the differences look too extreme. Magnitude -2.5 and magnitude -2.9 are both hella bright! Jupiter is always the brightest thing in the night sky, except for the Moon and Venus.

    1. I determined the size of the circles by taking -2.9 and -2.5 as the extremes of Jupiter’s brightness at opposition, choosing 2 and 0.5 millimeters as the largest and smallest circles, and proportioning the circle accordingly. It turned out to be the same subroutine I use to fix the size of star symbols in any plot; just a matter of deciding those numbers for different kinds of plot. (For example they have to be very different for a picture of the whole sky with only bright stars and a large-scale picture of Pluto among very dim stars.) In this case I’d have to choose different numbers if I was including stars in the picture, but what I want to do is exaggerate the variation in Jupiter, otherwise it wouldn’t be perceivable. Your comment is useful because it shows me I’ve exaggerated the difference too much and that’s confusing. All I have to do next time is change those millimeters 2 and 0.5 to say 2 and 1 and see how it looks.

      1. It’s taken me a couple of days to make time to sit down with my invaluable copy of Daniel Fleish and Julia Kregenow’s _A Student’s Guide to the Mathematics of Astronomy_ and do a little math and rough sketching. A difference of 0.4 magnitude corresponds to a brightness ratio of 1.445. If Jupiter at his brightest opposition is represented by a 2 mm circle, and at his dimmest opposition by a 1.4 mm circle, that looks about right to my eye.

        1. I’ve had to think a lot, while plotting of plotting hundreds of charts, about this sizing of symbols for magnitude, tried various formulae.
          Your suggestion of a difference matched to the actual brightness difference (which is (100^2)^(magnitudedifference)) is interesting and, in a way, obvious, and I’ll play with it sometime.
          But there may be no formula that fits all charts, because you want a range of sizes whose differences are perceptible to the eye, but that fit different ranges of stars. For instance one picture may include Venus and stars down to mag 6 – a range of 10 magnitudes – and you need a range of dot sizes from a large one for Venus to a tiny one for the faintest stars shown; but if you apply the same formulae to a picture featuring only the top twenty stars, a magnitude range of less than 3, you will have all large dots, unhelpfully similar to each other.
          So I continue my method of picking a high and low magnitude and a high and low size. Not necessarily “top and bottom” of either: if a star happens to be brighter or dimmer than the high and low magnitudes given, the equation will carry on up or down. A while back I settled on using 0 and 5 for the high and low magnitudes for most charts and worrying about only the two sizes, choosing those that had worked for previous similar charts or experimenting.
          According to a letter in Sky & Telescope, 1990 Nov., p. 462, by Peter A. Rogerson: “C J. Flannery showed in 1971 that perception of circles’ area is proportional not to radius squared but to radius to power 1.75; this constant is now widely used by cartographers.” (The number looks as if it is the inverse of a radian (57.3) divided by 100.) I made an effort to fit this in, but did not find a way to do so.

          1. At first I thought your response included a repetition typo, but then I realized that you must spend a lot of time plotting of plotting things. I appreciate all the charts, and my layperson’s musings are just my attempt to find a relation that made sense to me both quantitatively and visually in this one specific situation.

            Perhaps it’s only because I’ve grown accustomed to the illustrations in Sky and Telescope, but stellated icons seem to work well for representing very bright objects.

  2. That’s a great discussion and very informative plot of Jupiter’s possible opposition locations. Mars and Venus (possible inferior conjunction locations) would be really interesting too, given their inclination relative to the ecliptic. Not trying to give you more work or anything LOL! A collection of your diagrams and explanations of these sorts of things would really make a great Astronomy textbook.

    1. Thanks. I shall certainly be able to do likewise for Mars; it doesn’t have an opposition till 2016 May 22, but perhaps we can find a pretext. Its locus chart should be more interesting than Jupiter’s, since Mars departs more from the ecliptic, has more loop-like retrograde tracks, and varies more in magnitude. I don’t yet know how to calculate the locus for phenomena other than opposition, such as the greatest elongations or inferior conjunctions of Venus and Mercury, though a crude way would be just to calculate the positions of a lot of them.

      1. The crude way would be the only way I could do it :-) To switch back to the topic of Jupiter’s current opposition, I am having a nice view of the Not-so-great Pale spot here in Virginia on this evening of February 6 2015. I’m not a very skilled planetary observer but the spot seems fairly distinct tonight. What a pleasant consolation prize on an evening when the four satellites aren’t engaging in anything particular interesting.

      2. I could not resist the challenge of trying to show Mars’ locus of oppositions, so I decided to use the “work harder, not smarter” approach. The first four images at the above link (https://www.flickr.com/photos/starvergnuegen/) show the fairly accurate positions of all Mars oppositions from 1960 to 2054, an almost 100 year span starting around the time of my birth (1963). Unfortunately I did not scale the dots showing Mars to their magnitude or diameter, but I did capture that in the table I made with the data for those 45 oppositions, so perhaps I can try something in powerpoint to scale those orange circles. Guy’s chart will be much more professional for sure. There’s going to be some great Mars observing in 2018, 2020, 2033, and 2035!

  3. You were the first one to ever show me Jupiter, Guy. I look forward to our renewal tonight!

Leave a Reply to Anthony Barreiro Cancel reply