Science of Melancholia

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Science of Melancholia

#1  Postby NilsGLindgren » Jun 06, 2011 9:09 am

Hello,
Yesterday we saw Melancholia - a beautiful film about the end of the world. To put it concisely, an extrasolar planet falls from the interstellar void into the solar system, makes a close fly by about the sun, and apparently, passes close to the Earth (close enough to "take some of our atmosphere"), and first recedes, then the two planets make a head-on collision. It has not interacted with any other astronomical body. According to the film, the Melancholia has a diameter about four times that of Earth. One allegedly informed astronomically inteersted person makes the observation that the planet is receding at a speed of 100000 km/hr, so about 28 km/sec.
To me, this sounds wrong. If it falls from beyond the solar system, shouldn't it have the solar system escape velocity when it passes Earth?
Also, I am curious about the mechanics of the fly-by of a massive object. If, as would appear likely, Melancholia is a gas planet, it still would have the volume about 64 times that of Earth; with a density of 1 (like Uranus), the mass would be more than 10 times that of Earth. What would the mechanics look like in this situation? I speculated to what extent a close fly-by would cause a slingshot rather than a collision (not as spectacular in terms of FX, but quite deadly to life on Earth I would assume).
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Re: Science of Melancholia

#2  Postby Evolving » Jun 06, 2011 9:50 am

Just to address this one point:

NilsGLindgren wrote:If it falls from beyond the solar system, shouldn't it have the solar system escape velocity when it passes Earth?


I can see what you're thinking of, but the answer is "no".

If the object were to travel in a straight line towards the Sun, attracted by the Sun's gravity, yet somehow miss it and travel straight on beyond it, then at a first approximation it would have the same speed at each point behind the Sun as it had at the same distance when it was approaching it, and at the limit it would tend towards its escape speed: it would perform simple harmonic motion, like a spring. Even a spring, of course, in reality performs damped harmonic motion, because of friction, and so would our planet, because space is not empty (even ignoring its collision with the Sun itself!).

If the object travels into the Sun's gravitational field so as - initially - to miss the Sun, it would have a parabolic or - possibly - hyperbolic orbit around the Sun, and whether it escaped again from the Sun's gravitational field would depend on the velocity with which it entered that field in the first place, and on its interaction with the Sun and the remainder of the solar system - as you suggest, a slingshot effect might occur.



ETA:
at the limit it would tend towards its escape speed


Just to clarify what I mean by this: the magnitude of the escape velocity at any time depends on where the object is at that time: it reaches a maximum at the centre of mass of the solar system, and decreases as the object travels away from the Sun. So as our object performs its spring-like oscillation about the Sun (with the Sun somehow helpfully stepping aside each time it passes, without disturbing the field!), its velocity tends towards that maximum as it passes the centre of mass of the solar system, and decreases to become zero at the maximum extent of its amplitude.
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Re: Science of Melancholia

#3  Postby NilsGLindgren » Jun 06, 2011 10:08 am

OK, so whether the object that falls from outside the solar system collides with the sun or not is dependent on its initial direction and any interactions with solar system bodies, then? If it started "at rest" (this of course is a thought experiment), and did not interact with any other bodies, it would fall into the sun, I suppose? This would be the most simple variant.
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Re: Science of Melancholia

#4  Postby Evolving » Jun 06, 2011 10:14 am

Yes: if the object, initially, is travelling through the interstellar void, minding its own business, with a velocity only slightly different from that of the solar system (same direction, but very slightly slower), then as the solar system gradually overhauls it, it will be affected by the Sun's gravitational field and, on these (clearly unrealistic) assumptions, it will fall straight into the Sun with ever increasing speed relative to the Sun.
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Re: Science of Melancholia

#5  Postby NilsGLindgren » Jun 06, 2011 11:21 am

OK. So, in fact, we don't know with what velocity the object would be passing Earth, or, we settle for the figure mentioned in the film: 27 km/sec relative to Earth, and crossing the Earth orbit at close to 90 degrees.
How would the two objects interact?
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Re: Science of Melancholia

#6  Postby Evolving » Jun 06, 2011 11:49 am

OK, there is still one key fact that we don't know: how close the object approaches the earth (enough "to take some of our atmosphere", however close that is!). But if this is an object with, as you say, ten times the Earth's mass and crosses the Earth's orbit that close at a right angle, travelling more or less directly away from the Sun, then it would surely (without my having done any actual maths) knock the Earth out of its orbit (into a different, less circular orbit), away from the Sun, and that alone would be pretty bad news for life as we know it, without postulating a head-on collision later (after the object, presumably, has swung round the Sun one more time).
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Re: Science of Melancholia

#7  Postby NilsGLindgren » Jun 06, 2011 2:15 pm

As for the distance - I would discount the "take some of our atmosphere". The view on the closest approach in the fly-by gave an apparent diameter 5-6 times that of the full moon. Hard to judge, though, as it was close to the horizon. This causes me to underestimate. If it were 10 times that of the full moon, that would mean 5 degrees ... I tried to calculate it and got a distance of 45 Earth diameters, with the previous assumption that Melancholia has a diameter 4 times that of Earth.
That would mean something like 50 % more than the distance between Earth and the Moon. Oh dear, I have not done trigonometry since high school ...
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Re: Science of Melancholia

#8  Postby Evolving » Jun 06, 2011 2:45 pm

A back-of-the-envelope (literally!) calculation suggests to me that this is nowhere near close enough for Melancholia to take the Earth out of its solar orbit and put it into orbit around Melancholia instead, but it would surely disrupt the Earth's existing orbit significantly.

Bit hazy on the detail: haven't really done this celestial mechanics stuff beyond a fairly elementary level!

The figure you give for Melancholia's speed relative to the Earth is very close to the Earth's orbital speed around the Sun: so, as Melancholia crossed the Earth's path (assuming it crossed ahead of the Earth), they would be fast approaching each other during the manoeuvre.
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Re: Science of Melancholia

#9  Postby Evolving » Jun 10, 2011 1:05 pm

Interestingly (if you're that way inclined...) this is, as it were, a reverse slingshot effect.

The slingshot effect used - for instance - by the two Voyagers is like this. We have a large object (Jupiter, say) which is in orbit around the Sun, and rotates around its axis as it does so. A much smaller object (Voyager 2) approaches it, in the direction, more or less, of Jupiter's rotation, with a speed which is more than necessary to escape Jupiter's gravitational field, but not enough to escape that of the Sun, even at this distance. As Voyager passes Jupiter, it is attracted by the latter's gravitational field and its trajectory is bent around Jupiter into a partial orbit; as it does so, it picks up speed from Jupiter (Jupiter drags it along and speeds it up). (Relative to Jupiter, it gains kinetic energy and loses gravitational potential energy, at first, but once its trajectory starts to point away from Jupiter the reverse occurs.) Having gone part of the way around Jupiter, it escapes Jupiter's gravitational well with a speed - relative to Jupiter - identical to that with which it entered it: the gravitational well is symmetrical. Relative to the Sun, however, it has picked up a lot of Jupiter's speed, and its speed relative to the Sun now exceeds that necessary to escape the solar system.

The Melancholia thing is slingshot in reverse, because the object on its way away from the Sun is more massive than the one in a (stable) orbit. (Also, the two objects are far more similar in mass than Jupiter and Voyager!) But the important thing is how they are moving relative to each other. In that frame, the Earth is approaching Melancholia (as Melancholia crosses Earth's solar orbit) and it too is deflected, for the briefest of times, into an orbit around Melancholia. Like Voyager, Earth's speed is far greater than that necessary to escape Melancholia's gravitational field (as my back-of-an-envelope calculation indicated), but its solar orbit would be bent away from its current, almost circular one.
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Re: Science of Melancholia

#10  Postby NilsGLindgren » Jun 10, 2011 2:17 pm

I think I see. You have, by the way, in a most lucid way explained the mechanics of the sling shot effect :clap:
which I thought I understood until I read what you wrote and realized that I never did.
However, as concerns the Earth/Melancholia flyby, I would take your reasoning to indicate that a far more likely senario than a head on collision would be a disturbance of Earth's orbit (no laughing matter in and of itself)?
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Re: Science of Melancholia

#11  Postby Evolving » Jun 10, 2011 2:48 pm

A direct hit, so soon after Melancholia entered the solar system, would be a stupendously improbable event; and pretty improbable even after much time has passed, since there are far more possible orbits for Melancholia that do not intersect with Earth's orbit than those which do (let alone intersect with it such that both planets attempt to occupy the same space at once).

I suppose a deflection of the Earth's orbit is a lot less dramatic in the cinema than a collision!

Or it would make for a different kind of film: Earth subsides gradually into ice, followed by burning heat a few months later.
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Re: Science of Melancholia

#12  Postby Evolving » Jun 10, 2011 4:44 pm

Oh! Thanks for the kind words about my explanation, Nils.

I can do that, as long as the physics doesn't get too advanced! But I'm working on that...
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Re: Science of Melancholia

#13  Postby Evolving » Sep 22, 2011 8:58 pm

Hey, NilsG, if you get to read this: how did you manage to see Melancholia back in June? They have just started advertising for it here. In cinemas from 30th September.
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Re: Science of Melancholia

#14  Postby NilsGLindgren » Sep 23, 2011 5:08 am

Evolving wrote:Hey, NilsG, if you get to read this: how did you manage to see Melancholia back in June? They have just started advertising for it here. In cinemas from 30th September.

Simple - it started showing here, in Scandinavia, much earlier. After all, it's a Danish director, innit? It started showing in May.
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Re: Science of Melancholia

#15  Postby Evolving » Sep 23, 2011 8:47 am

Ah, makes sense.

Maybe they delayed its release here after his ill-advised comments at Cannes.
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Re: Science of Melancholia

#16  Postby NilsGLindgren » Sep 23, 2011 8:51 am

Possible. von Trier does not always display a great sense of judgement. "Do not start mouth unless brain has been switched on".
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Re: Science of Melancholia

#17  Postby klazmon » Sep 23, 2011 9:35 am

Friction can be ignored. It has no notable effect on the Earth's orbit over thousands of years. The scenario described requires the solar system capture the incoming planet. This can theoretically occur in a more than two body system (the reverse process of objects getting ejected) without an actual inelastic collision. The combined Sun - Jupiter system could capture and or eject objects which pass close enough to Jupiter on the right trajectory. The mechanics of it is utilised by spacecraft gravity assists to either get to more distant planets or in the case of the pioneer and voyager craft ultimately eject them from the solar system altogether. (One way you can guess that some of the gas giant moons are captured objects is that to get the reverse of a gravity assist they would end up in retrograde orbits). The chances of a planet sized object initially on a hyperbolic orbit being captured by the solar system without a direct hit on something on the first (and most likely only pass) sounds rather implausible to me. Much more dangerous and probable would be the close approach of another star. IIRC at least one relatively low mass star will pass by less than a light year from the solar system in the next few million years. If one passed close enough to disturb the Oort cloud we could be in trouble.

Oh yeah Gliese 710
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Re: Science of Melancholia

#18  Postby Khannea Suntzu » Nov 09, 2011 10:27 am

If Melancholia would enter from the Kuyper belt, drifting in with low relative motion, it would fall in and probably miss the Sun the first tumble. It would fall in with a very high relative motion, and considering its albedo those in charge of the planet would already know a year in advance, and would have started emigrating. Earth would have become a tyranny, where those wanting to leave and those wanting to make sure humanity (and life) would survive would be conspiring at all costs to make sure something would survive. A lunar or asteroid base would seem the only sensible idea, but I wonder if a year would be enough - even with all the world cooperating, to get this show on the road. A permanent base in a year would be extremely difficult. Five years yes - we would have a permanent settlement, one year 'iffy.

If the planet would have this trajectory I'd regard it as suspicious. Rather contrived - an entire planet! - but I'd regard it as a termination act. Destroy Earth and life completely. It's too unlikely, like hitting bullseye ten times in a row.

The orbit makes some sense, but the final stage doesn't. The movie is a cinematic statement depicting the frivolous irrelevance of human morons, especially highly entitled and useless rich human morons, and the planet is just a storytelling tool explaining how disinterested the cosmos is in our very elite humans.

The final stage of the impact misses realism in three ways -

* the planet would not 'capture' Earth at this impact velocity. If it went too far it would just 'merely' inflict a degree of Earthquakes and tidal upheaval (several kilometres of sea crashing over all continents) and we'd probably see the continents crack with volcanoes everywhere. The idea of an Earth 'captured' in a gravity wake of a Melancholia is like an Earth 'dislodged' from something like a 'proper orbit. Oh so Keplerian. Reality is Earth is a coalescled glob of warm fudge that would ripple and quiver like jellified roadkill with every passing truck on the highway. Melancholia's first pass would have turned Earth in to a sea of bright Magma, and humans would have started dying a week earlier.

But that would have left the cinematic point unmade - Humans right now are acting as children at best, irrelevant and self-centered idiots at worst.

And death can come to a world as easily as it comes to a single human life.

The best message of Melancholia is that would have done almost nothing if it were to happen. We'd cry and scream but we'd all die, and so what? It would be sad but nothing would be there to later on reflect on this manifest sadness. The tragedy exists only in ourselves and after the momentary pain and terror the Cosmos would barrel on more or less unperturbed and unimpressed. In fact the Cosmos as such doesn't even do 'impressed'.

The tragedy is we may be doing the same as Melancholia - in a century there might not be much life on this planet left, and certainly no human life. Would it be a greater tragedy if we caused an extinction of this magnitude ourselves? If your answer is yes - why do we not give a damn about even 10% chance of this happening?

The answer is - because we as individuals die. Why care about the abstraction of extinction if we ourselves all die? It isn't anyone's problem that the species may die 50 years from now. In the future we're all dead.
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Re: Science of Melancholia

#19  Postby kjordan » Dec 04, 2011 6:41 am

I just watched it tonight - very good movie, in terms of the psychological aspect (a depressive person becomes relatively strong and stable during a catastrophe, since catastrophes don't seem worse than normal life).

In terms of the science, I have several comments and questions - thanks for the information in this thread -

First, I don't remember the movie giving any significant info on Melancholia's size or composition. Was this info included in interviews or promotional materials? In the U.S. English version I watched, I do not think it was specifically addressed. Viewed through a telescope in the movie, I think I see clouds and water.

Second, the planet was described as being formerly "hidden behind the sun." I don't think it addressed whether the planet came from outside of the solar system; the impression I had was that it had always been on the opposite side of the sun from us, though perhaps they meant it had recently entered the solar system unnoticed because it did so on the other side of the sun. I don't know which possibility is more believable.

Third, it was described at one specific point as moving away from the earth at 60,000 miles per hour. That speed (relative to the earth) may not have applied at other points. The movie implies, but does not conclusively show, that the two planets do a "dance of death" after initially moving apart. This plays out in less than one day, during which Melancholia is always visible (larger than the moon), which seems unlikely to me; I would expect that to take more time to play out as the two planets are influenced by a common center of gravity.

Fourth, I realize the movie was intended to dig into psychology, not astronomy or physics. However, I found suspension of disbelief difficult to maintain, as Melancholia appeared to be the size of the moon one day, then enormously larger for the following two days. This would mean that Melancholia would play havoc with ocean tides the first day, and I speculate it would create massive earthquakes and volcanic episodes the second day. The third day, the earth would most likely deform and partially disintegrate as the two planets neared collision. The movie depicts surprise hailstorms and nervous horses, and terrified bugs, but an otherwise unchanged earth environment until the moment of collision. That moment is presented as the earth literally landing on the planet (sideways in relation to the characters), with a furnace blast of flame and debris rising from Melancholia toward the characters. I am sure the director didn't want to get bogged down in science, but I found the lack of science to be a distraction.

I recommend the movie despite any nitpicks (just don't expect Bruce Willis to save the day with a nuke).
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Re: Science of Melancholia

#20  Postby NilsGLindgren » Dec 04, 2011 10:08 am

kjordan wrote:First, I don't remember the movie giving any significant info on Melancholia's size or composition. Was this info included in interviews or promotional materials?

No. It was my off the cuff computation from the introductory scenes where the planet is seen close up to Earth. I further made the assumption that its compositon would be that of a gas giant like Neptune.
kjordan wrote:This would mean that Melancholia would play havoc with ocean tides the first day, and I speculate it would create massive earthquakes and volcanic episodes the second day.

Yes, agreed. My OP only adressed the first assumption, that of the collision.
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