An Inescapable Error

Contributed by
Nov 22, 2013
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Well, dangit. I screwed up.

This happens sometimes. What makes this one frustrating is that it’s one of those mistakes where I should’ve known what I wrote was wrong, but somehow got it in my head that it was fine. What’s worse is, I checked my math several times, but didn’t check my concept.

The mistake was in my article about Comet ISON yesterday. I said (in Fact #3) that the comet will be moving at an excess of 600 kilometers per second when it rounds the Sun on Nov. 28 next week. I calculated this by assuming the comet would be moving at the escape velocity for the Sun, which is the right way to do it… but I used the wrong number for the escape velocity.

The escape velocity of an object is how fast you need to move to escape from its gravity. Imagine you have a rock in your hand, and you gently toss it upwards. It goes up, slowing as it does, stops, then reverses course and falls back into your hand.

If you throw it harder — give it more energy — it moves faster, and goes up higher. A funny thing happens, though: As it gets higher, the gravity it feels from the Earth gets weaker. The force of gravity depends on your distance from it, so as the rock gets higher, it gets farther the Earth, and the force on it weakens.

Remember, though, the ball is slowing as it rises due to the force of gravity pulling it back. So if you balance things just so, give the ball just enough energy, get it moving at just the right speed, the amount it slows due to gravity will match the rate at which the force is weakening. It will always slow down as it gets farther from Earth, but will never quite reach zero. It will escape from the Earth, moving off (mathematically, at least) to an infinite distance.

The physics of this works in reverse, too. If you drop a rock from an infinite distance, it will be moving at escape velocity when it hits.

Physically, escape velocity depends on the mass of the object and its size. For the Earth, it’s about 11 km/sec (7 miles/sec). The Sun is much more massive, and has a far higher escape velocity: more than 600 km/sec.

However, that’s the escape velocity from the surface. If you start higher up, the escape velocity is smaller. That’s because gravity is weaker the higher you go, so you don’t need to move as quickly to escape. Comet ISON will pass about 1.8 million kilometers from the Sun’s center (the Sun is about 700,000 km in radius, so it’ll pass about 1.1 million km over the surface). If you put that distance in for the starting point, the escape velocity is about 360 km/sec (225 miles/sec).

That’s still incredibly fast, far faster than any man-made spacecraft has ever traveled. It’s 1500 times faster than a commercial jet, and fast enough to cross the United States in less than 15 seconds! It’s also about 0.1% of the speed of light.

So it won’t be moving quite as quickly as I originally wrote, but it’ll still be screaming fast.

And while I’m at it, in that same article (Fact #12) I wrote that after it passes the Sun, ISON will be visible in the west after sunset. I should’ve been more careful writing that; it will be visible in the west after sunset, but it will also be visible in the east before sunrise, like it is now! That’s because it will be moving north after it rounds the Sun, actually passing near Polaris in January. As it heads in that direction, it moves into the part of the sky that neither rises nor sets (that is, moves above or below the horizon line), but instead circles the pole endlessly. Stars (and in this case, a comet) in that region of the sky are called circumpolar, and are visible all night long.

So right after perihelion — its closest approach to the Sun — weirdly, the comet will be visible in the east in the morning and the west in the evening. For my latitude (40° north) it’s actually a bit higher off the horizon in the morning. So if ISON survives its close encounter with the Sun, I’ll be up early a few days later to see it for myself. I haven’t seen it yet (clouds plus my sinuses have kept me from getting up early enough to try), but I’m hoping to.

I hope you do too, and remember that making mistakes are a part of life. The best thing to do is own up to them… and then use them as an excuse to talk about physics. Yeah, I think that’s for the best.

[My thanks to several BABloggees who pointed this out to me. Crowd sourcing is pretty good at spotting errors like this!]