it seems like forever ago now

Our planet is estimated to be about 4.54 billion years old. There are lots of cool things that have happened since that fateful day — scientists believe it was a Tuesday, but the most religious types insist that it was a Monday. The truth may never be known.

We've been waiting for a while now... but for what???More interesting to me, is the idea that a year hasn’t always been a year long.

We may not be able to discover which day of the week our planet was thrown together, but it brings up a series of questions—whose winding logic I will spare you from—that lead to this one big question: was a year in 2012 the same length as a year in 4,450,000,000 BC? In other words, how has the earth’s orbit changed since it was began circling round the sun?

This is going to be an intense calculation, so for this entry, I am just going to list what I think may be the factors most important.

In theory, it’s fairly straightforward to find a body’s orbital period P. If we know the semi-major axis, a, and the mass of the two bodies in question — m1 and m2 — we can use Kepler’s formulations to discover that

P = 2πa * [a/G(m1 + m2)]1/2

Using standard units for all of the constants above, the mass of the sun, the mass of the earth, and the distance of earth’s semi-major orbital axis, we get P = 3.156 x 107 seconds, or about 365.3 days, which is pretty close to the SI standard of the Julian year, defined as 365.25 days of 86400 SI seconds each. That approximation gets us to within 0.01% (or nearly 1 part in 10,000). That’s pretty decent, so we’ll go with it.

Luckily there aren’t many variables in this equation, but let’s look at each. First up, the mass of the earth. Sadly, the sun is about 333,000 times more massive than the earth and so any mass variation that may have occurred in our planet’s long history—even if it is as high as a 10% change in mass—is completely overshadowed by the effects of that massive burning ball of hydrogen we call a sun.

And what about that sun? It’s sending out a crap-ton (excuse the jargon) of energy each year, which has been converted by reactions taking place deep within the sun’s core (think E = mc2). It’s also losing protons and electron constantly as solar winds. So yeah, we hit the jackpot.

According to Wikipedia:

The early Sun had much higher mass loss rates than at present, so, realistically, it may have lost anywhere from 1–7% of its total mass over the course of its main sequence lifetime.

And finally, the semi-major axis of our planet’s orbit. If we assume a 1% rate of loss of mass by the sun over 5 billion years, the distance changes by about 75,000 kilometers (source), which is about 0.05% of the distance from Sun to Earth. According to Kepler’s Third Law (P2 ∝ a3), this distance scales as a third power, while the orbiting period scales as a square. Therefore these changes in the semi-major axis become negligible very quickly. So let’s go ahead and neglect them.

Wow, ok. Now let’s calculate the period P using the sun’s “error bars” we talked about for its mass (I put “error bars” in quotes because I know some wise statistician is going to read this and flip his or her lid and then I will get fired from the internet for misusing stats terms — my apologies, God of stats).

To use the middle of the approximation given above for the sun’s changing mass, we’ll assume that it has lost 4% of its total mass since 4.54 billion years ago. Sparing you the calculation, the initial mass of sun, Mo, is approximately (1/0.96)Ms, or 1.04167Ms, where Ms is the current mass of the sun.

We’ll denote the change in the period as ΔP. There are a million steps next, and they involve things like Kepler’s Third Law, at least 3 cold ones, and the quadratic formula. Ultimately, we find that ΔP = -637,600 seconds — or 7 days, 9 hours, 6 minutes, 40 seconds less than the length of a standard year.

Therefore, the first earth year ever was only 356.7 days long. That must’ve been hard having no Christmas that year … awww, poor kids. 😦

Whew, that was cool. One more tiny thing to take care of — how many “years” old is the universe? Or more simply put: how many times has our planet made the trip around our sun?

Finding the average length of the year and adding it all up is pretty straightforward, and so i’ll leave the plug and chug deets for the chumps (that would be me). In the end we find that the earth has orbited the sun some 4.494 billion times, a mere 98.99% of what the lame-stream media would have you believe …