Asteroid Apophosis, Orbit Changes, and Boy (not)Geniuses

You might have heard the story that’s been going round about the asteroid
Apophis.
This is an asteroid that was, briefly, considered by NASA to be a collision risk with earth. But after more observations to gather enough data to compute its orbit more precisely, the result was that it’s not a significant risk. The current NASA estimates are that it’s a collision risk of about one in 45,000.

The news around it is that some German kid claims to have figured out that
NASA got it wrong, and that the real risk is 1 in 450. What was NASA’s big
mistake, according to the kid?

He says that if the asteroid were to hit a satellite, that it would change the satellite’s trajectory enough to make it hit the earth.

This has been reported with ridiculous credulity. Anyone with the least
bit of mathematical literacy should know, pretty much without even needing to
think about it, that this is absolutely silly.

According to the estimates that I’ve been able to find, the mass of
Apophosis is about 1.2×1011 kilograms. That’s 120 billion kilograms.

The mass of a Boeing communication satellite (which from a few minutes of
searching around the net appears to be average) is about 1400 kilograms.

The asteroid is around 100 million times larger than a satellite.

Let’s put that into perspective. A pretty average car – a Toyota Corolla – has a mass of roughly 1400 kilograms. A common housefly has a mass of 16 milligrams. That’s pretty close to the same ratio of masses as the asteroid and a typical satellite. So when we talk about how much the trajectory of the
asteroid could be changed by hitting a satellite – we’re talking about
roughly the same kind of thing as how much a bug can alter the trajectory of a moving car by crashing into its windshield.

Obviously, the speed of the collision matters. But still – take a fly moving at 1,000 kilometer per second, and collide it with a car, and the
change in speed of the car is going to be miniscule: the momentum
of the fly at that velocity is 1×106m/s × 1.6×10-6kg = 1.6 kg m/sec; in a perfect collision,
the maximum change in velocity of a 1400 kilogram car would be around
1 millimeter per second. If the fly were moving at one million kilometers per second – which is quite a bit faster than the speed of light! – it could change the velocity of the car by a meter per second. (Originally, I screwed up my units here, and got the speed of light wrong by a teeny little bit – like three orders of magnitude. D’oh.)

So how much could a collision with a satellite alter the orbit of an
asteroid? Not bloody much. How much would it need to alter the course of the satellite to make it hit the earth? A lot. A huge amount. A satellite impact would be occurring at a range of around 6 times the radius of the earth (based on the closest approach distance of the asteroid.) The asteroid would be following a trajectory very close to tangential to the earth’s surface at this point. So to turn it to a collision would require a dramatic shift.

There’s just no way to make this work. It’s impossible. And anyone with a bit of common sense should be able to work this out. Not necessarily to this degree of detail – but for goodness sake, the fact that a huge rock in space isn’t going to have its path altered significantly by hitting a tiny little tin can should be bloody well obvious.

45 thoughts on “Asteroid Apophosis, Orbit Changes, and Boy (not)Geniuses”

1. Erik R.

Bravo. I saw this story, too, and, without thinking about it much, there was a vague taste of bullsh1t to it. People so yearn for the underdog that they’re willing to swallow a story like this.
Good work, as usual.

2. mds

One million kilometers per second is indeed a significant fraction of the speed of light. In fact, it’s over three times greater than it.

3. Mark C. Chu-Carroll

mds:
Yeah, you’re right… For some reason, I was thinking
the speed of light was 3×108 kilometers per second, but that’s meters per second, not kilometers. Stupid, stupid mistake on my part.

4. Anonymous

So to turn it to a collision would require a dramatic shift.
The original claim was that it turns its next orbital pass into a collision. But still unlikely to happen. Especially given that it will miss the satellites anyway.

5. Anonymous

“He says that if the asteroid were to hit a satellite, that it would change the satellite’s trajectory enough to make it hit the earth.”
Don’t you mean the asteroid’s trajectory here?

6. Nathan

Obviously, a satellite doesn’t have enough momentum to cause the asteroid to make a right turn on a dime and smack into the earth, I think even the German kid could understand that. I think what he was going for was that if the asteroid hit a satellite on its first encounter Earth in 2029 it could peturb the asteroids orbit a teeny bit, enough to cause it to hit the Earth when it makes another encounter in 2036. I’m sure you know that many-body gravitational dynamics is inherently chaotic and differences in the initial conditions lead to solutions which tend to diverge from each other exponentially fast. In this light, the boy’s assertions seem to be a tiny bit more logical. The certainty with which the news report quotes the 1 in 450 chance is a little troubling, and I doubt a 13 year old school boy would have the necessary mathematical knowledge to calculate these odds or integrate the equations of motion. This makes me doubt the report, just not for the reasons given in your blog post.

7. ice9

It may be worth mentioning here that apophasis is a rhetorical device in which a speaker pointedly passes something over as a way of mentioning it slyly, as in
I need not mention NASA’s many computational errors.
ice

8. Aureola Nominee, FCD

I’m sorry to add yet another correction, but…
the speed of light in a vacuum is approx 3×10^9 meters per second, not 3×10^8.

9. Nathan

Having typed that number into a calculator far too many times, I can say with certainty that it’s definitely 3E8 m/s.

10. dhawk

On a somewhat related note, the article also mentioned the prediction that it would hit in the Atlantic ocean. Some have defended this, which seems ludicrous to me. If we are so uncertain in its trajectory that we cannot rule out an impact, how could we possibly know where it would hit?
The argument was that we know where it would come from, and how long it will take to get here, and we know what side of the earth will be facing that direction, so the prediction of the Atlantic ocean isn’t as outlandish as it seems. It would require that we know how long it would take to get here to very great precision.
So that got me thinking about the errors. I think that small perturbations in the orbit propagate in x^2 for distance (the relevant figure for the Atlantic ocean prediction), but in x for projected position. Therefore, it may be meaningful to say, there’s only a 1 in 45000 chance that it will hit us, but if it does, it will likely hit in the Atlantic (or on that side of the Earth).
Any thoughts on this?

11. Anonymous

I’m sorry to add yet another correction, but…
the speed of light in a vacuum is approx 3×10^9 meters per second, not 3×10^8.

No, it’s definitely 299,798,456 m/s ~ 3*10^8 m/s. That’s one of the few constants I bothered to memorize.

12. Matthew L.

@Nathan,
It’s true that any perturbations will add up to large effects in many-body celestial mechanics problems, but the two problems with the kid’s claims are that there are much larger perturbations involved than colliding with a man made satellite, so such effects probably already fall inside the error ranges being made (note they say 1:45000, not zero). The second problem is that although it’s fair to describe the divergence between perturbed solutions as exponential, in celestial mechanics they still tend to be a long time by human lights. Radioactive decay is exponential too, but sometimes with extremely long half lifes.

13. Doub

According to the article below, the boy didn’t claim that touching a satellite would make it go onto the earth directly, he predicted that hitting a satellite when coming close to the earth in 2029 could alter its path and increase its chance to crash on earth in 2036 during its next orbit.
http://www.physorg.com/news127499715.html

14. Harshblogger

Not that it matters but a Boeing 702 communications satellite has a dry mass (mass without fuel) of ~2500 kg. With fuel at launch the mass can be ~5000 kg.

15. Thomas

Regardless of whether or not the effect is significant, you have to credit the boy for coming up with a real complication that it is possible no one had thought about before. (Or mabye the experts had but quickly realized it was insignificant). That is quite different from the “girl genius” who supposedly debunked global warming but in reality only parrot stuff fron contrarian websites.

16. Odysseus

Agree with Thomas. You shouldn’t just blame the boy for claiming a higher collision risk in 2036. The major part of his work is a calculation of the asteroid hitting anything at all in 2029, including satellites. Here he comes up with a collision risk of 1/450. I don’t know how he came to this result nor wether this is correct, but if it is that’s quite remarkable an effort for a 13-year-old. He did not, as far as I know, say much about a possible change of the trajectory or even try to correct the risk of an impact in 2036. This did of course not matter to the journalists who apparently wanted to come up with a “killer asteroid” although the focus of the boys’ work was the probability of the event “asteroid hits a satellite” rather than “we will all die”. Suppose there has been a bit of mis-framing going on here 🙂

17. Daithi

I tend to agree with Odysseus. I think it is pretty cool that a 13-year-old boy tackled a problem like this even if he was wrong. The news media on the other hand deserves condemnation for being idiots. They publish sensational claims by 13-year-old boys without doing some fact checking.
When I was 13 I solved one of the world’s most famous math problems – how to trisect an arbitrary angle in geometry. As it turns out, I trisected an arbitrary triangle and not an angle, but just working on this problem taught me more about geometry than any math class I ever had. So even if the kid got it wrong I still applaud the effort, and I have no doubt he knows more about science than the vast majority of kids his age.
However, the ‘professional’ journalist deserve a flogging. Hopefully this kid won’t feel ridiculed and turn sour on exploring questions like this.

18. jackd

Using Mark’s figure of a velocity change of 1mm/sec, I get a position change of 220km over seven years. I don’t know if I’m figuring this right, but if I am, then it looks like it’s about two orders of magnitude from being a significant effect on the chance of a collision with earth. Heck, I’d bet the change is well within the error bars of our measurement of the original velocity.

19. Mark C. Chu-Carroll

I don’t care whether this kid was talking about an encounter on this orbit, or on the next pass, or fifty passes from now.
The fact is, the effect of impacting a satellite would be so miniscule due to the mass difference that there it’s lost in the noise.
N-body gravitational problems are incredibly complex. So being able to say, absolutely where the rock is going to be 30 years from now is extremely hard. But the potential change caused by hitting a satellite is absolutely miniscule in comparison to the other factors that affect it. It’s just lost in the noise of all the other uncertainties.
Remember – we don’t even know the exact mass of this rock; we can just estimate it based on its observed orbital path. Think about that – the amount of uncertainty about the mass of the asteroid is orders of magnitude larger that the mass of any satellite it could impact.

20. QrazyQat

On a somewhat related note, the article also mentioned the prediction that it would hit in the Atlantic ocean.
As if that wouldn’t be a problem.

21. ColoRambler

The fact is, the effect of impacting a satellite would be so miniscule due to the mass difference that there it’s lost in the noise.

Although the current story about collisions with satellites is nonsense, people have seriously proposed deflection scenarios based on slamming a large spacecraft into the asteroid to give it a well-defined velocity change.
This is only workable well before the 2029 close encounter, for a variety of reasons. The main reason: leading up to the 2029 close pass, there is only a very narrow range of possible trajectories (sometimes called a “keyhole”) that actually leads to an impact in 2036 or later. As a result, only a small change in velocity is needed to ensure a miss. 10-100 km over a decade (or so) before 2029 is enough to do the job, and a spacecraft impact could easily accomplish that. To be sure, a number of very precise position and velocity measurements must be made before such a mission could hope to succeed (some people have proposed sticking a radio transponder on the asteroid).
The “keyhole” exists right around closest encounter, so a direct hit with a geosynchronous satellite can’t possibly change things — there’d be far too little position change in the available time. Nevertheless, under the right circumstances the basic concept (slam something into the asteroid to make a difference over a decently long time) is plausible.

22. iwdw

The more interesting problem isn’t deflecting that thing, it’s capturing it.
If any significant portion of that rock is iron, the value of a source of building materials in orbit would be incalculable to any government that could do it.

23. Mark C. Chu-Carroll

iwdw:
It’s a nice dream, but it doesn’t really work out.
It’s far from incalculable. It’s 120 billion kilos of rock. You can work out an approximate iron content, and use that to work out the value of the material. Then, from that, subtract the cost of propellant for the amount of acceleration needed to move it into orbit. Then subtract the cost of building an in-orbit refinery that can process tons of metal in zero-grav.
Economically, it doesn’t work out. It’s way the hell to big: we don’t need 100 billion kilograms of iron for any foreseeable orbital project – and it would only be when we wanted that kind of volume that it would make any sense. For the amount of metals we need for space construction for the foreseeable future, it’s far cheaper to shoot hunks of iron into space on rockets.

24. KeithB

What about the moon? That at least has a reasonable delta V compared to the Earth. Remember “The Moon is a Harsh Mistress”?

25. Anonymous

What the little boy is talking about is that a collision with a satellite in 2029 could change enough the apophosis´ trayectory to make it impact the earth in 2036 (nor in 2029).
Its not a pool balls problem, it is a gravitational problem.
Still dont know if the little boy correction is correct.

26. dileffante

Off topic: I don’t remember having seen obits here, but perhaps you could post a bit about the Lorenz attractor and why Edward Lorenz findings were so important. “…brought about one of the most dramatic changes in mankind’s view of nature since Sir Isaac Newton” are quite strong words!

27. joe

I saw a response by either NASA or the ESA (I forget which), and they stated that while the asteroid will come closer than the geostationary sattelites, the trajectory does not intersect these orbits. By the time the asteroid crosses Earth’s equatorial plane (which is where most of the sattelites are), it will be much lower than the sattelites. It seems to me that this was indeed factored into the calculations.
I also wondered about the ability of a 13 year old to do these calculations. Even if he had done the calculations by plugging the numbers into a commercial software package, the ability to interpret the output in reaching the 450:1 conclusion would be far beyond any 13 year old’s ability (except maybe someone like Gauss).

28. Michael Welford

Hmmm. If Apophis collides with a satellite 100 million times less massive than itself and that satellite gouges out a crater ejecting about 100 times its own mass, then we might suppose that Apophis’ speed is reduced by about 1 part in a million. If Apophis travels several billion miles between 2029 and 2037 and its speed is changed by about 1 part in a million then we might expect its location in 2037 to be a few thousand miles different from what it would be without the satellite collision.
Uh-oh!
I just hope we have the good sense over the next 21 years not to put a satellite in Apophis’ path.

29. Nick

The satellite in question (assuming one actually existed in Apophosis’ path, and apparently none will be) wouldn’t have to deflect it enough so it would hit Earth on this path, but it would have to deflect it enough so that it could make it into that “keyhole”, which seems improbable (though this won’t be for another 11 years, so I suppose its theoretically possible someone will send up a couple billion kilogram satellite in an unusual orbit between now and then).
But the biggest problem with this story isn’t that a 13 year old screwed up a math problem (hell if I was called out for each error I made in math class in junior high, well lets just say the media wouldn’t have had as much time to cover the OJ trial). The problem is that the original stories were claiming scientists had confirmed the kid’s math, which turned out to be completely false.

30. magetoo

#25-26:
Ah, wild speculation, my favourite. 🙂
I’m thinking that the better option would be to land a (self-repairing/replicating) autonomous mining operation on the asteroid. Adjusting the orbit is going to be too expensive, and/or take too long time — but if you could build things “on site” and shoot them off on trajectories that would lead to them being captured by Earth later, that might be worth it. And some form of active capture seems more plausible when what you’re trying to move is not absolutely enormous.
Too bad that the orbit doesn’t seem to be too useful, at least from a quick reading of Wikipedia. (i.e. it doesn’t get you a free ride to Mars or somewhere else fun and easily colonizable)
I’d also like to suggest that by the time we need 100 billion kilograms of iron it would make sense to have already started changing the orbit decades ago. (But that still won’t make any sense, if you have the option of not changing it.)

31. ferfuracious

“It may be worth mentioning here that apophasis is a rhetorical device in which a speaker pointedly passes something over as a way of mentioning it slyly, as in
I need not mention NASA’s many computational errors.”
If you want something better sounding than apophasis, give praeteritio a try.

32. Wry Mouth

Of course, you have neglected reports of the NASA/Boeing joint project to launch a communications satellite in 2010 which is made entirely of matter ejected from neutron star supernovae. NASA has been busy collecting this matter since the 60’s, it is believed, beginning with the Mercury program. Such a satellite, in addition to being nigh-invulnerable and capable of sub-atomic super-conduction in the near-freezing and/or boiling vacuum of space, would weight in the neighborhood of 110 billion kilograms [emphasis mine, using HTML tags for style!].
Should this project ever be realized, then we would certainly all be in danger, despite the increased clarity and reliability of T-Mobile service in certain parts of Utah.
;o/

33. g

I keep reading “apophosis” as “apoptosis”. Asteroid apoptosis might be really useful.
(But it’s actually “Apophis”; Mark, you should probably fix the title even though it will wreck everyone’s puns on “apophosis”.)

34. ianam

“I’m sure you know that many-body gravitational dynamics is inherently chaotic and differences in the initial conditions lead to solutions which tend to diverge from each other exponentially fast.”
I’m sure you’re wrong; these are not chaotic systems.

35. ianam

I don’t know how he came to this result nor wether this is correct, but if it is that’s quite remarkable an effort for a 13-year-old.
If [false] then [anything].

36. ianam

you have to credit the boy for coming up with a real complication that it is possible no one had thought about before. (Or mabye the experts had but quickly realized it was insignificant).
Gee, ya think? So we don’t have to give him credit for coming up with something blatantly obvious … except to folks like you, apparently.

37. Tim G

You need to be careful when you drive because if you hit a tsetse fly, you could go spinning out of control.

38. Jonathan Vos Post

The JPL trajectory effort I spoke with about this last week was not aware of the popular press (mis)reporting. I was not familiar with his NASA insider sources.
He asked me if I knew about the “keyhole.”
I asked if he meant a collision with a KH-11 satellite?
No, he told me. The keyhole in orbital phase space. There is some complicated manifold issue. I asked him to email me a clarification for this blog, but he’s busy in the vicinity of our nation’s capital this week, evaluating NSF research proposals. If and when I hear back from him, I’ll try to report here.

39. Peter Gerdes

Actually figuring out where it would hit if it hits (like the atlantic) is pretty easy. It’s hard to figure out whether to will impact the earth or miss by thousands of miles but we can figure out when it will pass by/hit the earth and look at what part of the earth will be facing towards the direction the asteroid is arriving from.
Also the boy’s story as reported in the news has always struck me as ridiculous. I mean in order to by this story you have to accept that hitting a satellite radically increases the chances of this asteroid hitting the earth. How could we know that unless we had a precise enough understanding of it’s path to give a better probability in the first place.
A more plausible version of what the kid really did is that he calculated the chance the asteroid impacts an object under earth’s gravitational control. That’s nice and I’m sure this is a smart kid who probably will deserve a scholarship in 5 years but there’s no reason this needs to play in the nes.
BTW having known enough of these ‘genius’ kids my primary reaction to these situations is to feel sorry for them. Yes, they can make significant accomplishments but (having been in this position to some degree myself and knowing others who had it worse) many of these accomplishments happen because the kids throw themselves into work when their personal lives aren’t going well. It doesn’t make the accomplishment any less and often this is the best response to a bad situation but I still feel bad.