Monday 2 December 2019

Lunar Perigee Syzygy

This article is prompted by a question from @SophieJane96x [1], viz. was the Moon closer on Sunday night because it looked bigger than normal.

Well, the short answer to the question is that actually the Moon was rather farther away on Sunday than it's mean distance. Specifically, the mean Earth-Moon separation is about 385,000 km (that's ca. 239,200 miles for those who like old money, or about 79,733 leagues or so for those who like really old money [2]). At noon on Sunday 1st December it was [3] 396,003 km (246,065 miles, 82,022 leagues, ...) away which is 11,003 km [4] further away.

That being said, the actual answer is, as always with these things, a bit more complex. Because I can imagine when Sophie was looking on Sunday night the moon did appear larger.

But first we need to do a little bit of celestial mechanics.

Also, I promise you that syzygy is a real word, and not a bizarre typo.

The Earth-Moon System

The Moon orbits the Earth [5] on an almost (but not quite!) circular orbit. Technically speaking, it has a mean eccentricity of 0.0549, although due to the interaction of the Earth-Moon-Sun system it can vary between 0.0266 and 0.0762 [6]. By way of comparison, the Earth's orbit around the Sun has an eccentricity of about 0.0167 so the Lunar orbit is rather less circular than the Earth's around the Sun.

An elliptical orbit means the Earth-Moon distance is constantly varying. It transpires that the complex interaction of the Earth-Moon-Sun gravitational system [7] is actually really well modelled. In addition, it also transpires we can measure the distance to the moon at any time to about ±2 mm [8]. Or to put it in political terms, we are more certain where the Moon is than we are what voters think about Brexit.

Because we astronomers like giving names to things, we call the time when the moon is closest to the Earth perigee and when it is furthest away apogee [9]. Exactly how far away the moon is at each of those points varies depending on what the eccentricity is at that time, which you can see in this graph by Darekk2 that I have scurrilously nicked from Wikimedia Commons

If the Moon's distance from the Earth is varying, it stands to reason that how big it appears on the sky (formally: what angle it subtends) will vary too. And you would be correct! The difference in apparent size [10] between apogee and perigee is about 14 %. Again, I have viciously stolen an image from the Commons (this time made by Tomuren) to illustrate this

The important thing to bear in mind here is that while that difference looks like it is obvious, without a reference to compare against it really isn't.

So, yes the apparent size of the Moon does vary due to its orbit. But not by all that much at all.

And, by the way, everything here applies notwithstanding the phase of the Moon. Last night was a waxing crescent according to the widget on my phone [11]

As an aside (because it has nothing to do with size) the Moon is very noticeably brighter at perigee compared to apogee. Why is left as an exercise for the reader (hint, one over r-squared ;-) ).

Syzygy and Supermoons

Ok, so what are these 'supermoon' things we see in the press every now and then? And what's that weird vowel-free word?

Well, first off. No real astronomer will use the term supermoon, if only because it's ill-defined. The term was coined by an astrologer [12] [13] to do whatever it is astrologers do. The idea was that a full moon which coincided (or nearly coincided, or whatever) with perigee is in some sense marked and special.

Now, it transpires astronomers do have a name for the exact moment of a full moon. That's that syzygy word. It is basically any orbital configuration where three things are in a line — it comes from the Greek meaning 'yoked together'. Not just the Earth-Moon-Sun, it equally could apply to Sun-Mercury-Mars (which has been observed!). The exact moment of a 100 % full moon is by definition a syzygy. So, if this moment was to also exactly coincide with Lunar perigee, then you would have a syzygy-perigee.

(A new Moon is also a syzygy, but this time the order is Moon-Earth-Sun, and is why Solar [C1] eclipses can only occur at new Moons)

Which would be spectacular in the sense of being an unusual event (the woolier astrological definition of supermoon occurs a few time a year), and it would be (as hinted above) brighter than your garden variety full moon. And it would be 14 % bigger than the 'average' one.

But that's probably still not why anyone would have thought the Moon was big last night. After all, last night was neither perigee (it's only four days from apogee, actually), and it clearly wasn't a syzygy.

And just for the avoidance of doubt: the Moon being at perigee, or syzygy, or in Ares, or none of these makes no difference whatsoever. Save that obviously a less full Moon causes darker nights. That bits real (but astrologers don't seem to talk about it, oddly)

The Moon Illusion

Apparently the ancient astronomers knew about this phenomenon, and it was also recounted by Immanuel Kant. But what is it?

Broadly speaking, all things being equal, an observer perceives the Moon, when it is low in the sky (i.e. close to the horizon) as being larger than when at zenith (the highest up it gets). As the name suggests, this is entirely an illusion, a fact which one can trivially confirm with a camera. The apparent size of the moon doesn't perceptibly change over the course of a night [14].

We astronomers may have measured the distance to the moon to within two millimetres, but the psychology of human perception is not really our thing. So why this happens is a little bit unclear.

According to Wikipedia (which I trust here because, quite frankly, I know nothing more than "this thing happens"), there are a few competing ideas. The first seems to be called the "apparent distance hypothesis" [15] To explain this, let's imagine a cloud approaching from the horizon [16] (in daylight so we can see it for the moment). When it is just over the horizon, the cloud, no matter its actual size, appears small. As it approaches and rises overhead, it appears to get bigger (it's apparent size increases), reaching a maximum overhead (at zenith) before then decreasing in size as it retreats. 

The argument then goes that the human brain is so used to this phenomena, that when an object like the Moon — which as we have established subtends the same angle all night — moves across the sky, the brain assumes it must larger on the horizon and smaller at zenith. Just like a cloud would behave. [17]

An alternative explanation is the "relative size hypothesis". This says, simply, that when low on the horizon one has lots of reference points (all of which are fairly small) to compare the Moon against, whereas at zenith it is surrounding by a whole lot of nothing. Consequently one perceives the size as changing. 

Personally I like the first explanation. Either way, the take-home message here is that if you see the Moon in the sky and think "gosh it's big", it's almost certainly an illusion.

Why is it red?

Why are sunsets sometimes red? No, seriously, you're asking the same question. Now, Rayleigh scattering is a bit beyond this blog post, but that's the key thing. There is bit of an inversion between an 'ordinary' moon low in the sky and a lunar eclipse, which we'll come to in a moment.

The ultra simple explanation is that as light passes through the atmosphere it is scattered, and the more it goes through (there's some stuff about angles here that I am entirely ignoring ) the more it is scattered. So blue light scatters first (hence the sky is blue), and red light last (so that's what's left when everything else is gone) [18].

So, when the Moon is low in the sky, like the Sun at sunset the light which reflects off it then has to pass through an awful lot of atmosphere (at a steep angle, but we'll park that) to get to you, hence red.

Now, for a lunar eclipse — which, by the way, is an example of a syzygy! — the Moon is always going to be pretty high in the sky, so why is it red? Well, when the Moon is perfectly in the Earth's shadow, the only light hitting it comes via the atmosphere, and is therefore already red. It then reflects and comes back down to you.

Magic! And none of this psuedo-scientific 'blood moon' nonsense.

What about 'Harvest Moons'?

Just a name for the full moon closest to the Autumnal Equinox. Nothing special about it all.

Or 'blue moons'?

Aside from some esoteric corner cases mostly relating to things like volcanic eruptions, which we shall pretend don't exist, this has nothing to do with the colour of the Moon.

The Lunar cycle is ~28 days long (actually it is a little less but who cares), which obviously means that although most months will have one full Moon, and most seasons three; some must have two and four respectively (another way of thinking about this is that some years must have thirteen full moons).

These 'extra' full Moons are called blue moons, and hence the idiomatic phrase "once in a blue moon" to denote an unlikely [19] event.

Or 'black moons'?

The internet tells me this is something to do with having an extra new Moon in a month, but it seems like a modern invention, so let's ignore it and let it die out.


[C1] Let he who is without error point out the first erratum, yes I originally wrote 'Lunar' here because I wasn't paying attention.


I always said I would write something about astronomy, though admittedly I didn't think it would be this.

Hopefully it all made (some?) sense.

[1] Whom I only mention because it will annoy her. Sorry-not-sorry.
[2] Which I make to be 824 million cubits, or about 2 million stadia, or... I'll stop now.
[3] Is there anything Wolfram Alpha doesn't know?
[4] That 3 isn't a typo, though really I should just round it down, it isn't adding anything.
[5] There are probably conspiracy theorists who disagree, they have the disadvantage of not having a PhD in Theoretical Astronomy, and are therefore wrong.
[6] Buried somewhere on this webpage.
[7] And other increasingly exotic stuff like radiation pressure, tidal dissipation, the very small effect of the outer planets, ...
[8] I didn't believe this either when I first read it, but it is true, as this paper nicely explains (Reasenberg, R.D.; et al. "Modeling and Analysis of the APOLLO Lunar Laser Ranging Data".)
[9] These are the two Greek terms (we like Greek) for the apsides of the Moon's orbit around the Earth. The suffix -gee means we're talking about the Earth, then apo- or peri- refer to furthest away or closest, respectively. Cf. perihelion (cloest to the Sun), apocytherion (furthest from Venus), perigalacticon, apochron, ... Anyone caught sticking Greek prefixes on Latin stems will be summarily banned from all astronomical activities.
[10] We say apparent because, obviously, the actual size of the Moon doesn't change.
[11] Of course I have a widget on my phone for this.
[12] Astrology is not a science; astronomy is. Don't insult astronomers by calling them astrologers; please do insult astrologers by calling them charlatans, for they are. (I mean don't really, that's mean, but you get the idea)
[13] If you really must I'm told you can read the original article. Disclaimer: I haven't even clicked this link to check it works, let alone read it.
[14] I say perceptibly intentionally because it is slightly changing due to the elliptical orbit discussed above. But even if you could discern the 14 % apogee-perigee difference, you wouldn't discern the night-to-night difference. So we can approximate it to "doesn't change".
[15] Which Wikipedia attributes to Cleomedes, who is otherwise famous for writing an early astronomy textbook in AD 200, which also is the earliest source for Erastothenes' measurement of the size of the Earth.
[16] This also works with æroplanes, birds, UFOs, very long distance golf balls, cruise missiles, and indeed just about anything that flies.
[17] Some wily soul is going to point out here that the Moon is indeed very slightly closer to the observer at zenith than on the horizon; however, that effect is tiny (I've checked, changes of 1,000 km in six hours are not unusual) compared to the Earth-Moon separation change overnight; which we have already established is imperceptible. Clouds are much lower down, so there the effect is very noticeable.
[18] Real astronomers: yes I know I've left loads of things out, no don't shoot me, I'm trying to explain this without writing a thesis.
[19] I mean, blue moons are not all that unlikely, but idioms be idioms!

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