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.