The Gravity Thief

(I'm reading The Quantum Thief right now, so I figure vague, sciencey-sounding titles are appropriate.)

My good friend (read: successful and well-liked blogger on whose posts I sometimes leave inane comments) Phil Plait recently blogged about the impending non-Armageddon that is 2012 DA14. The brief summary is that a giant hunk of rock half the size of a football field will just barely miss us this Saturday, and astronomers are so excited they're polishing their mirrors in anticipation. For a slightly more nuanced explanation, check out the Bad Astronomer's blog.

What intrigued me about Plait's post is the minor detail that, currently, 2012 DA14 has an orbital period of 366 days, but after its interaction with the Earth its orbital period will shrink to 317 days. This means, obviously, that the asteroid will go around the sun more quickly from now on. But why? Is it speeding up, or just getting closer to the sun? The answer is yes.

Four hundred years ago, after pouring through Tycho Brahe's enormously detailed astronomical records, Johannes Kepler devised three empirical laws of planetary motion. The relevant one here is the third law, which states that the square of a body's orbital period is proportional to the cube of its semi-major axis (radius). So the closer a planet is to the sun, the shorter its period.

Newton later confirmed Kepler's observation with his universal law of gravitation, which states that the force between two objects is proportional to the product of their masses divided by the square of the distance between them. So, then, if an orbiting object's period decreases, its radius does also, and if the radius decreases, its speed increases.

We know from the kinetic energy formula that a faster object has more energy, which means that 2012 DA14 will gain energy from its interaction with the Earth. How much energy? Well, using Kepler's third law, we can see that if the orbital period shrinks by 49 days, the semi-major axis will shrink by about 9%.

With a bit of calculus, we can figure out that Newton's law of universal gravitation predicts that the gravitational potential energy of an object is inversely proportional to the distance between it and another object. So, a 9% reduction in radius is a 10% increase in energy. (Because gravitational potential energy is defined to be negative, this can be thought of as gaining negative potential energy, or losing positive potential energy. The end result, however, is an increase in kinetic energy).

And energy, as we know, is conserved. So if the asteroid is gaining energy from its encounter with the Earth, the Earth must be losing energy. It looks as if the asteroid has a mass of about 130 million kg, which is less than the mass of the Earth by a factor of 46 quadrillion. Thus, a 10% increase in energy for the asteroid is a 2.4x10^-17 % decrease for the Earth. And if we go backward through the math, we see that the Earth's orbital period will increase by about 1 nanosecond. Put another way, after 28 years, an extra 1 second will have elapsed when compared to the previous 28 years (all else being equal).

The all else being equal part, however, is quite a stickler. The duration of the Earth's orbital period and day vary quite significantly due to effects from the moon, the sun, earthquakes, glaciers, and a whole range of other factors. It's likely that more than just the asteroid's speed changes during its near-Earth encounter, and the same goes for the Earth. Rather than our orbit changing, it might alter the length of the day by a tiny fraction.

But there is an underlying principle here: energy conservation. Energy never disappears completely; it just moves from one object to another, changing forms as it does so.

Don't turn that dial!

Okay, so this is a little late, but I want to talk about my physics lab from last Wednesday. I haven't had a lab course since I took Chemistry my junior year of high school, and that was eleven years ago. Working with a lab partner, writing down results, adjusting the apparatus -- these are things I (until recently) thought I was probably done with. Since I'm trying to become a scientist, however, I may have to get used to this routine.

Anyway, the lab was split into two parts. The first part had me and a partner measuring the spring constant of a spring by pulling a force gauge attached to said spring. After recording the force from the gauge and the distance we'd stretched the spring, we could find the slope of a Force vs. Distance graph, and that slope was the spring constant in newtons/meter -- the higher the value, the stronger the spring.

The exercise was decidedly unrelated to electricity and magnetism (unless you want to talk about the fact that chemical bonds are electric in nature). I think the point was just to acquaint us with doing a lab, which I suppose is necessary given that some of the students haven't taken a lab course in eleven years.

After that, however, the professor had each of us individually play with an analog oscilloscope. We weren't attempting to measure anything with the oscilloscope -- just push buttons and turn dials so that we could familiarize ourselves with its function.

For those unfamiliar with an oscilloscope (I had heard of but never seen one before last Wednesday), it's a device that measures an incoming electrical signal and displays a corresponding Volts vs. Seconds graph.

Here's a picture of the model we were using:

Manipulating and calibrating the o-scope (as the prof called it) is a somewhat tricky enterprise. The display is divided into a grid, and you can adjust the number of volts and seconds per length of grid. As you can see, however, there are a wide variety of other knobs and controls that are required to produce a clean image of the incoming signal. And throughout this whole process, I was wondering, is this really necessary?

I mean, perhaps I'm putting the cart before the horse, but this is 2013; shouldn't there be an iPhone app that can do this for me? (Yes.) In fact, isn't that exactly what any piece of audio equipment does when it translates electric signals into sound -- measure the frequency and amplitude of the wave?

Yeah, I'm in an introductory E&M course. Yeah, we're going over the basics. But is learning to fine-tune an o-scope like a WW2 radio operator really going to be useful in our later physics careers?

I can imagine two ways in which it might be. The first possibility is that while of course there will be software that can identify the sinusoidal wave (or whatever) of an electric signal when we're working scientists, someone has to write that software. In that case, having an intuitive feel for what we're attempting to measure might be useful.

The other possibility I can imagine is that a working scientist will have to play with significantly more complex pieces of machinery, and training on a somewhat antiquated oscilloscope is a necessary first step toward that goal. I mean, you can't start with the LHC, right?

I thought of both these counter-arguments to my original complaint, but I don't know how much stock I put in either of them. Unless I'm actually going to use an oscilloscope as a scientist, isn't there something that more closely resembles a piece of modern scientific equipment that we could be using?

According to wiki:

Oscilloscopes are used in the sciences, medicine, engineering, and telecommunications industry. General-purpose instruments are used for maintenance of electronic equipment and laboratory work. Special-purpose oscilloscopes may be used for such purposes as analyzing an automotive ignition system, or to display the waveform of the heartbeat as an electrocardiogram.

Ah, I did not know that EKGs were essentially specialized o-scopes. That's neat! But I'm a little hazy on what "maintenance of electronic equipment and laboratory work" means for general-purpose o-scopes. A little more reading tells me that they can be used to test the changing voltage of a device to make sure it's working within expected parameters. That's obviously useful, but again I feel as if this is something that could be done by a piece of software. Do we really need a human to turn a dial until the display stops spazzing, or can we just use a do...while loop?

Anywho, I think that's enough griping for now. Perhaps I should be a theorist.

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(I'll probably post something substantive later today.)