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Episode 108: Practical Application of Uncertainty - Getting Around the Solar System

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Recap: Episode 94 was probably kinda boring, learning about error, uncertainty, accuracy, precision, and all that stuff. Well, it's back, but this time more interesting! Learn all about what our inability to make perfect measurements means for how we get around the solar system - from Superman flying to Mars to satellites orbiting to asteroids hitting Earth to observing objects with spacecraft.

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Introduction: Since there's no specific claim to be talked about in this episode, I'm going to give an introduction based on a dream I had recurring when I was ... youngER. Imagine, if you will, that you're flying high above the planet, soaring around the globe like Superman. And then it's time to go home. But, you're not sure where it is. You kinda know the general area, and maybe you can get close, but who knows where specifically you need to go?

Now let's say you're Superman and are flying to Mars, and you're blind. You start off in the general direction you're told, but what if you're off in your trajectory by the width of a human hair. On Earth, staring at Mars, the width of a human hair is nothing. But propagate that more than 60 million kilometers and it could be an issue.

What this hypothetical child's dream gets to is the topic of this episode, and it's something I only learned about recently.

Setup

People like to think they know things. I know where my car is. Just as important, I know where my car keys are. I know where my keyboard is as I wrote this episode. I can touch-type, so I know where the keys are. If I'm off by a little bit, my fingers can easily slide to where they're supposed to be.

I know where the refrigerator door is. If I'm particularly tired, then I may miss the handle at first, but I get progressively closer with each try and finally can find it.

When I have my stalker spy camera lens on my camera - a 400mm with 2x extender, and I aim towards the moon, I start by putting the lens in the general direction of the moon, and then I fine-tune the pointing, then I can usually see at least a little bit in the viewfinder, and then I fine-tune it more and I'm finally aimed at the moon exactly. Since even with an 800mm effective lens on a 1.6x crop factor sensor for a nice 1280mm equivalent focal length still doesn't have the moon completely fill the field of view, I don't have to be pointed exactly at the moon, I have a little bit of leeway.

Gravity: Satellites

But what if we're flying a satellite? Sure, you have the Rocket Equation - something that Mike Bara thinks Werner von Braun stuck an extra term into to account for hyperdimensional physics and no one noticed - and you can use that along with orbital dynamics and Newton's form of Kepler's laws to figure out where your satellite is going to be. But how well can you predict that? How good are our numbers? If we leave that satellite alone for a year, will it be exactly where we thought it would be in the absence of any other outside force?

The answer is "sort of," and that is mainly because we don't know very well how strong gravity is. Gravity is among the worst-constrained fundamental forces. Many of you have probably heard of the Gravitational Constant, or "big-G" for short. This was a term that Newton introduced in his universal law of gravity, and it's a number that can be thought of as how strong gravity is. Many people have measured it over the years, though the first measurement wasn't until 150 years after Newton's death, by Henry Cavendish just before 1800.

Cavendish measured it by setting two large balls near two smaller balls, the two smaller ones held in place by a suspension system, with the idea being that the two large would attract the two small, twisting the suspension, and you could measure the time it took to twist a certain amount and the amount of force required to twist it and derive big-G. The whole thing was kept in a wooden box to minimize air currents and temperature variations, and since this was 1797-98, there were no big trucks going by. For the time, it was an incredibly accurate measurement and he got it to within about 1% of todays' measurements, an accuracy that wasn't surpassed for a century.

And yet, with all our modern equipment, modern experimental design, and everything else, we only know it to 0.012%: (6.67384±0.00080) x10^-11 m^3/(kg*s^2).

This might seem highly accurate. 0.012% relative uncertainty is miniscule. But, let's go back to being Superman aiming for Mars. From Earth, being off by just 0.012% in your initial trajectory can cause you to miss Mars by a huge amount.

Similarly, we have to know big-G when calculating orbits and trajectories. In the last episode, I talked about forces and gravity, and you kinda have to know the AMOUNT of that gravitational force that will be pulling you in any direction to calculate your orbit.

Over the short-term, like a day or a week or a month, 0.012% uncertainty in that force is not going to amount to much. But, it adds up.

Gravity and Other Forces: Asteroids

As another way of looking at this, let's talk about asteroids, and the likelihood of them hitting Earth. Here, you not only don't know gravity to an arbitrary precision, but you also don't know exactly where it is when you see it.

Think of it this way: You're looking at a satellite image of the United States or whatever country you want, on your computer screen. Unlike Superman, you have studied geography and know where your house is. But it doesn't matter. It doesn't matter how exactly you know where your house is on that map, you are staring at a finite number of pixels on your computer screen. You can click on that pixel that represents 100 km on a side and then zoom in, and you may be close. But now you need to click on a pixel that's 50 km across and zoom in again. 25. 10. 5. 1. And by this point, when you're looking at pixels that are more like 10s of meters or yards on a side, you can click on the few that are your house.

Now zoom back out to the 100 km per pixel map, and try to figure out how far it is from your house to, say, the middle of Toronto Canada. Because each pixel is 100 km on a side, it's going to be hard to make this estimate at the 1-km level.

You might think this is a contrived example, but it's not: This is exactly what observational astronomers have to do when observing asteroids and trying to calculate their trajectories. Let's say you observe an asteroid that you know from past observations is about the distance of Mars, and Mars is at its closest approach to Earth, about 60 million km away. Your field of view is that of a survey telescope, maybe, say, 1° on a side. And you're using a very expensive high-resolution astronomical CCD array, 4096 pixels on a side (about 16 Mpx). You manage perfect seeing because you took the picture with a camera on a satellite, unblurred by Earth's atmosphere. How big is each pixel?

You can use trigonometry to figure it out: Each pixel corresponds to very roughly 250 km (400 miles) on a side. You know the asteroid is only a few hundred meters on a side, meaning that even though it should be just a tiny 1/1000th the size of each pixel, you are limited to the size of that pixel.

You DON'T know where that asteroid is to any accuracy better than 250 km. And I gave you really ideal circumstances and a very high-resolution CCD in this scenario -- it's usually worse than that.

But, let's say you use orbital dynamics equations to figure out where it should be tomorrow night. It's still out by Mars in terms of distance, so each pixel is again 250 km. But you predict that it should move 30 pixels, or about 7,500 km. Plus or minus some uncertainty. It has moved 35 pixels. Knowing its new location, you are able to project a revised orbit and lower the uncertainty on its orbit. This is why all new asteroids initially have very unknown orbits, but more and more observations let us narrow down the uncertainty.

Now let's say you observe it as it passes by Earth's moon, at the distance of Earth's moon. You use your same telescope and camera, and that means that each pixel now is only 1.6 km (1 mile) across. You're still not going to resolve the asteroid, it will still be 1 pixel, but you now know its position to much MUCH higher accuracy.

Circling back to gravity, combine several observations of asteroids that are farther away than Mars, with our 0.012% uncertainty on the value of big-G (remember, that's how much gravity actually pulls on things), and you get the point where you may start to understand how odds of impacts are calculated for certain asteroids to hit Earth. They're based on uncertainty ellipses. We think we know where its trajectory will take it, but projecting that 1 year into the future results in a certain uncertainty to any given confidence level.

Meaning that we can say we're 50% certain it will be within a very narrow region of space, but 99% certain it will be within a somewhat larger region of space. 99.9999999999% certain it will still be in the solar system. That kind of thing.

If instead you want to project it 10 years instead of 1 year, then either your error ellipse increases in size, or your confidence has to drop. So you can be 50% sure it will be in a broad region, but only 10% sure it will be in a very narrow region of space. More observations, higher-resolution observations, closer observations, a better value for big-G, and you can better predict its orbit.

Practically speaking, that's what happened with asteroid (99942) Apophis. When it was first discovered in 2004 and we had only a few observations, its probability of hitting Earth 25 years later in 2029 was calculated at 2.7%. Just like we thought that asteroid would move 30 pixels in a day. But, just as we observed it to move 35 pixels instead, observations of Apophis just in that day rose to 64, and the chance of it hitting Earth was raised to 1.6%. Two days later, we refined its diameter and impact probability to 2.2%. Two days later, on December 27, 2004, observations were 176 and the probability of Earth impact was 2.7%. Just later that day, after calculating its orbit BACKWARDS in time and looking at archival images where it should have been - meaning we had a longer baseline in time of 287 days instead of just 5 and so could narrow the uncertainty - probability of a 2029 impact dropped to 0.004%.

It dropped that much because those previous estimates, though based on 176 observations, had only been over the course of 4-5 days. Just like if you were Superman headed to Mars and only adjusted your trajectory in the first five seconds of your trip. But, the archival data gave us so much more time to extend the analysis backwards that we could refine the orbit much better, like Superman instead adjusting his path in the first 5 minutes instead of 5 seconds.

As observations continued over the years, the orbit got better and better pinned down. 2029 was ruled out for an impact, as was 2036, though 2053 was still better than 1 in a million that it would hit. When Apophis came close to Earth in 2013, we got a better measure of its physical parameters and orbit and so could make even more accurate predictions.

As of May 2013, based on 13 radar observations, 7 Doppler observations, 3987 optical observations, all spanning 3318 days, the chances of it hitting before 2060 are less than something like 1 in a billion, which is our threshold for saying that it WON'T hit. The highest chance of it hitting is in April 2068, with the odds at 1 in 256,000, meaning it has a 99.99961% chance of MISSING Earth.

To complicate things, other than exactly where it is and gravity, exactly how it spins and other effects like the pressure of light are poorly understood. At this level, when we're trying to project the orbit out 50 or 100 years, these kinds of tiny effects can add up, just like our uncertainty in the value of big-G, and contribute to RAISING the uncertainty in our estimates.

I like this example overall because it's a very good way of showing the entire nature of science: It's an iterative process. We can never really know what IS going to happen, but the point is to get progressively closer to that objective observation that will show us whether we were right or wrong with our models.

Observing Pluto

As a final example of the practical application of uncertainty in this episode, I'm going to talk about Pluto. Some of you may know that I've been doing some work helping to plan the observations that the New Horizons spacecraft will make as it approaches, passes by, and retreats from Pluto next year - closest approach is July 2015.

In planning these, there are many things that have to be taken into account. Stuff like the amount of memory on the craft, the amount of fuel, how much space each observation from each image takes in memory, how quickly it can be downlinked back to Earth and how often we can do that, which telescope on the Deep Space Network will actually be pointed towards Pluto at the time.

And, where Pluto actually is, and where its moons are.

We don't know how big Pluto is. We don't know where it is. We don't know how big its moons are. We don't know where they are or where they will be.

Sure, I'm exaggerating a bit if I were to use every-day definitions of "know." We have a gagillion observations of Pluto and its main moon, Charon. We have many observations of Nix, Hydra, Kerberos, and Styx. But, we don't have enough. We don't have a long baseline of time, especially for the four ones that were discovered in the last decade - Nix, Hydra, Kerberos, and Styx. In fact, we still call Kerberos and Styx P4 and P5 for planning purposes because they didn't have names while we were planning observations.

We have a decent idea of their orbits. We have so many star occultations by Pluto that we know its radius is 1,184 km, but the uncertainty on that is ±10 km, or about 0.84%.

That means that when we want to plan an observation with the LORRI instrument, the main camera, and we want to image Pluto's limb - the edge of the disk - we have to take that 0.84% uncertainty of its radius into account. And uncertainty in its orbit. It means that when we point the narrow slit of the spectrometer of Alice and plan for it to lie across Hydra, we have to take into account our limited knowledge of exactly where it will be when that observation is going to be made.

How much uncertainty do we account for? Just like if I'm taking my 1280mm camera lens and aiming it at the moon, let's say I didn't know where it was to better than 200% its size. And, I can't look through the viewfinder live and correct for that -- Pluto is over 4 hours light-travel-time away! So that means that if I wanted to point where I think Hydra is going to be, I may only have a 70% chance of it being in the field of view of my camera, because I don't know where it will be precisely, or even how big it is.

Is a 70% chance good enough for this once-in-a-lifetime observation on a spacecraft that is going to take nearly a decade to get to Pluto, traveling faster than a bullet?

On the team, we say, "no." We plan for a 2.5-sigma observation, meaning that we need to be 98.8% sure we're going to get the object in our observations. That means instead of one image, or one spectrum, or one observation in general, we may need to take two. Or three. With slightly different pointings of the spacecraft to make sure that we cover that 98.8% probability ellipse. That means each object we want to observe will take up more storage space and require more time to downlink to Earth. Which means we actually can't do quite as much science as we want ... but, that's the price we pay for making sure that we actually observe the object we want to observe.

And, since a few of you are really into this stuff, I'll add that as New Horizons approaches Pluto, it will be talking images of the system several times a day, and the team will be using those data almost live to update where the moons and dwarf planet itself will be. In our planning now, we take into account an assumed updated accuracy for the orbits.

Wrap-Up

What does all this mean? Well, I think Episode 94 wasn't that interesting, The Difference Between Error and Uncertainty. It was a lot of dry stuff. But it's an INCREDIBLY important topic, and the ideas contained within it apply to pretty much everything. In contrast, my discussions on image analysis around episodes 47 and 48 WERE well-received, even though, at their root, they are talking about the same kinds of things -- what you can actually get out of your data, and what those limitations are.

In that spirit of practical application, I've focused on three real-life examples of how those concepts of uncertainty play out in our solar system, how our limits on measuring fundamental constants to the universe contribute to our limited accuracy in the ability to predict even where things are going to be.

Pseudoscientists like to think that we know everything to an arbitrary uncertainty. And so then when scientists come out with a revised number, they scream that science is a failure because they've said the old stuff is wrong. Alternatively, pseudoscientists or prognosticators like to say that scientists are only willing to put qualified probabilities onto numbers, values, or whatnot, while THEY are saying with certainty something is or is not so or will or will not happen.

These all show a basic lack of understanding of how science works. And hopefully now, after this episode, you will be better equipped to understand it yourself.

Provide Your Comments:

Comments to date: 1. Page 1 of 1.

David   Washington Metra Area

1:31pm on Saturday, May 3rd, 2014

Tropisms can be far more accurate than calculated trajectories. Most animals and plants rely on tropisms, not planned trajectories. Heat seeking missiles use a tropism not a calculation. If the stimulus is directional, the source of the stimulus can be easily found by following the gradient of the stimulus.

I read a Superman comic where something like this happened. He was blind, on earth, and flying into the sun. I hypothesize that he was using a heat-seeking tropism.


It happened like this. One villain planned an escape as follows. He had a large vat of liquid lead read with some type of molten lead squirter. He covered Superman with lead head to toe and then escaped.

Superman can not see through lead even with his X-ray vision. In those days, he could not even melt the lead with his heat vision. The rules have changed since that time. However, he could not melt the lead off with X-rays.

What he did was fly into the sun and back. The heat of the su... read more »

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