What If? (32 page)

Sparta

Q.
In the movie
300
they shoot arrows up into the sky and they seemingly blot out the sun. Is this possible, and how many arrows would it take?
—Anna Newell

A.
It’s pretty hard to
make this work.

Attempt 1

Longbow archers can fire eight to ten arrows per minute. In physics terms, a longbow archer is an arrow generator with a frequency of 150 millihertz.

Each arrow spends only a few seconds in the air. If an arrow’s average time over the battlefield is three seconds, then about 50 percent of all archers have arrows in the air at any given time.

Each arrow intercepts about 40 cm
2
of sunlight. Since archers have arrows in the air only half the time, each blocks an average of 20 cm
2
of sunlight.

If the archers are packed in
rows, with two archers per meter and a row every meter and a half, and the archer battery is 20 rows (30 meters) deep, then for every meter of width . . .

. . . there will be 18 arrows in the air.

18 arrows will block only about 0.1 percent of the Sun from the firing range. We need to improve on this.

Attempt 2

First, we can pack the archers more tightly. If they stand with the density of a mosh pit crowd,
1
we can triple the number of archers per square foot. Sure, it will make firing awkward, but I’m sure they can figure it out.

We can expand the depth of the
firing column to 60 meters.
Th
at gives us a density of 130 archers per meter.

How fast can they fire?

In the extended edition of the 2001 film
Lord of the Rings:
Th
e Fellowship of the Ring,
there’s a scene where a group of orcs
2
charge at Legolas, and Legolas draws and fires arrows in rapid succession, felling the attackers with one shot each before they reach him.

Th
e actor playing
Legolas, Orlando Bloom, couldn’t really fire arrows that quickly. He was actually dry-firing an empty bow; the arrows were added using
CGI
. Since this fire rate appeared, to the audience, to be impressively fast but not physically implausible, it provides a convenient upper limit for our calculations.

Let’s assume we can train our archers to replicate Legolas’s fire rate of seven arrows in
eight seconds.

In that case, our column of archers (firing an impossible 339 arrows per meter) will still block out only 1.56 percent of the sunlight passing through them.

Attempt 3

Let’s dispense with the bows entirely and give our archers arrow-firing Gatling bows. If they can fire 70 arrows per second, that adds up to 110 square meters of arrows per 100 square meters of battlefield!
Perfect.

But there’s a problem. Even though the arrows have a total cross-sectional area of 100 meters, some of them shadow each other.

Th
e formula for the fraction of ground coverage by a large number of arrows, some of which overlap each other, is this:

With 110 square meters of arrows, you’ll cover only two-thirds of the battlefield. Since our eyes judge brightness on a logarithmic scale, reducing the Sun’s brightness to a third of its normal value will be seen as a slight dimming; certainly not “blotting it out.”

With an even more unrealistic fire rate, we could make it work. If the guns release 300 arrows per second, they
would block out 99 percent of the sunlight reaching the battlefield.

But there’s an easier way.

Attempt 4

We’ve been making the implicit assumption that the Sun is directly overhead.
Th
at’s certainly what the movie shows. But perhaps the famous boast was based on a plan to attack at dawn.

If the Sun were low on the eastern horizon, and the archers were firing north, then the light
could have to pass through the entire column of arrows, potentially multiplying the shadow effect a thousandfold.

Of course, the arrows wouldn’t be aimed anywhere near the enemy soldiers. But, to be fair, all they said was that their arrows would blot out the Sun.
Th
ey never said anything about
hitting
anyone.

And who knows; maybe, against the right enemy, that’s all they need.

Drain the Oceans

Q.
How quickly would the oceans drain if a circular portal 10 meters in radius leading into space were created at the bottom of Challenger Deep, the deepest spot in the ocean? How would the Earth change as the water was being drained?
—Ted M

A.
I want to get
one thing out of the way first:

According to my rough calculations, if an aircraft carrier sank and got
stuck against the drain, the pressure would easily be enough to fold it up and suck it through. Cooool.

Just how far away is this portal? If we put it near the Earth, the ocean would just fall back down into the atmosphere. As it fell, it would heat up and turn to steam, which would condense and fall right back into the ocean as rain.
Th
e energy input into the atmosphere alone would also wreak
all kinds of havoc with our climate, as would the huge clouds of high-altitude steam.

So let’s put the ocean-dumping portal far away
—
say, on Mars. (In fact, I vote we put it directly above the Curiosity rover; that way, it will finally have incontrovertible evidence of liquid water on Mars’s surface.)

What happens to the Earth?

Not much. It would actually take hundreds of thousands
of years for the ocean to drain.

Even though the opening is wider than a basketball court, and the water is forced through at incredible speeds, the oceans are
huge
. When you started, the water level would drop by less than a centimeter per day.

Th
ere wouldn’t even be a cool whirlpool at the surface
—
the opening is too small and the ocean is too deep. (It’s the same reason you don’t get
a whirlpool in the bathtub until the water is more than halfway drained.)

But let’s suppose we speed up the draining by opening more drains,
1
so the water level starts to drop more quickly.

Let’s take a look at how the map would change.

Here’s how it looks at the start:

Th
is is a Plate Carrée projection (c.f. xkcd.com/977).

And here’s the map after the oceans drop 50 meters:

It’s pretty similar, but there are a few small changes. Sri Lanka, New Guinea, Great Britain, Java, and Borneo are now connected to their neighbors.

And after 2000 years of trying to hold back the sea, the Netherlands are finally high and dry. No longer living with the constant threat of a cataclysmic flood, they’re free to turn their energies toward outward expansion.
Th
ey
immediately spread out and claim the newly exposed land.

When the sea level reaches (minus) 100 meters, a huge new island off the coast of Nova Scotia is exposed
—
the former site of the Grand Banks.

You may start to notice something odd: Not all the seas are shrinking.
Th
e Black Sea, for example, shrinks only a little, then stops.

Th
is is because these bodies are no longer connected to the ocean. As the water level falls, some
basins cut off from the drain in the Pacific. Depending on the details of the sea floor, the flow of water out of the basin might carve a deeper channel, allowing it to continue to flow out. But most of them will eventually become landlocked and stop draining.

At 200 meters, the map is starting to look weird. New islands are appearing. Indonesia is a big blob.
Th
e Netherlands now control much of Europe.

Japan is now an isthmus connecting the Korean peninsula with Russia. New Zealand gains new islands.
Th
e Netherlands expand north.

New Zealand grows dramatically.
Th
e Arctic Ocean is cut off and its water level stops falling.
Th
e Netherlands cross the new land bridge into North America.

Th
e sea has dropped by 2 kilometers. New islands are popping up left and right.
Th
e Caribbean Sea and the Gulf of Mexico are losing their connections with the Atlantic. I don’t even know
what
New Zealand is doing.

At 3 kilometers, many of the peaks of the mid-ocean ridge
—
the world’s long-est mountain range
—
break the surface. Vast swaths of rugged new land emerge.

By this point, most of the major oceans have become disconnected and stopped draining.
Th
e exact locations and sizes of the various inland seas are hard to predict; this is only a rough estimate.