Today we have a solution to our killer asteroid challenge

episode, but before we get to that

there's something very special about today.

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November 25, 2015 is the hundredth birthday

of space-time.

Not the show, the fabric of the universe.

On this day in 1915, Albert Einstein

first presented his complete theory of general relativity

to the Prussian Academy of Sciences

in Berlin, completely changing the way

we understand the cosmos.

Now, we cover a lot of material on this show,

but Einstein's beautiful theory is

an essential part of what inspires us here

at "Space Time."

General relativity's profound description of space and time,

of matter and energy, emerged from the simplest

of thought experiments, simple statements about reality

and yet the mathematical description

that flowers from those statements

describes our universe with stunning accuracy.

The elegance of this theory has inspired so many students

of physics to follow in Einstein's path exploring

the mysteries of the universe.

It inspires us at "Space Time" to try to share those mysteries

with you.

In order to keep doing that as best we possibly can and also

so that our friends and family don't kill us

for missing Thanksgiving, we won't be releasing an episode

next week.

Instead, we'll be working hard on what's to come.

On December 9, we'll delve deeper than ever

into the weirdness of black holes, after which we'll

start exploring the nature of matter and time.

But for now we have some math to do

and it's all going to be Newtonian.

Newton was OK, too, I guess.

In our recent challenge episode, I

asked whether you could save the Earth from a killer asteroid.

In a hypothetical scenario, the asteroid Apophis

will buzz by the Earth in 2029 but then hit the Earth in 2036.

You were tasked with assessing the plausibility

of saving the Earth with a gravitational tractor.

You'll pulsed fusion drive spacecraft

is to intercept the asteroid in the 2029 pass

and pull Apophis 25,000 kilometers ahead of its

would be location using only the gravitational attraction

of the spacecraft.

Let's solve this in two steps.

First, some Newtonian mechanics to see

how much mass is needed to accelerate Apophis

to get the desired change in position

and then the rocket equation to see how much fuel we'll

need to achieve that.

To make the mechanics easy, we can simplify some stuff.

First, let's do the calculation in the frame

of reference of Apophis' 2029 orbital velocity.

That way we don't need to worry about the asteroid's initial 30

kilometer per second velocity at all.

In the asteroid's frame, its starting velocity is zero.

We're also just going to figure this out in one dimension.

The x-axis is the orbital path of the asteroid.

We just need to add enough velocity to Apophis

to move it 25,000 kilometers relative

to the frame of reference of its initial motion.

The gravitational force between the spacecraft and Apophis

is providing all of the acceleration and hence velocity

change needed to pull Apophis that extra distance.

How much mass do we need to provide

the necessary acceleration?

Well, we know that the spacecraft's mass

is changing because it needs to burn fuel to accelerate.

But we can just calculate the average acceleration

based on the average mass of the spacecraft

plus fuel over the seven years.

That average acceleration will give us

the same change in velocity as we'd

get using calculus to determine the acceleration at every point

based on the changing mass.

So the mass that we're going to calculate

is the average mass of the spacecraft

in fuel over the seven years.

We need to use one of the basic kinematic equations, the one

relating change in position, average acceleration, and time,

with a starting velocity of zero in the frame of Apophis

in 2029.

Using this equation, the average acceleration

comes out to around 10 to the power

of minus 9 meters per second squared

or around one ten billionth of Earth's surface gravity.

What mass is needed to produce this acceleration?

Our spacecraft is hovering 325 meters

from the center of mass of Apophis.

Newton's law of universal gravitation

tells us the force between two massive objects.

It also gives us the acceleration

experienced by Apophis due to the mass

of the spacecraft and the fuel.

Note that Apophis' mass cancels out, but don't worry.

That 30 billion kilograms will come

back later when we need to figure out

how much fuel we need.

Note also the shape and composition of Apophis

doesn't matter at all for this part of the calculation, which

is a big part of the advantage of the gravitational tractor.

So the average mass needed to achieve

the acceleration we calculated earlier

comes to 1,600 metric tons.

This is around 80% of the mass of the space shuttle

so we can definitely do this.

But what about the fuel requirements?

For that we need the rocket equation.

It tells us the relationship between delta v,

the total change in velocity, to the exhaust velocity

of the fuel and the ratio of fueled to unfueled

or wet to dry spacecraft mass.

But what are these masses?

We're trying to pull the entire asteroid,

so we have to include its mass as part of the spacecraft mass.

But the mass of Apophis is enormous

compared to the spacecraft-- 30 billion kilograms

compared to the 1.6 million kilograms we got earlier.

So the dry mass may as well just be Apophis' mass.

The wet mass is then just Apophis' mass

plus the fuel mess.

Rearranging all of this, we get this equation

for the ratio of fuel mass to asteroid mass.

OK, but what about the exhaust velocity?

We know that our pulsed fusion drive has an exhaust velocity

of 500 kilometers per second.

However, remember that the thrusters have

to be angled to miss the asteroid,

otherwise we'll just push the extra backwards.

We'll just calculate that angle assuming Apophis

is a 325-meter diameter sphere.

The angle of the thrusters ends up being around 30 degrees,

so the effective exhaust velocity

is the backwards or reverse component of the thrust.

The sideways components end up being useless.

Note, however, that you have to have thrusters pointing

in at least two directions on either side of the asteroid

to cancel the sideways component out.

The reverse component of the exhaust velocity

ends up being 427 kilometers per second.

Putting this together, we get a ratio of fuel mass

to asteroid mass of 5.3 by 10 to the minus 7.

So the fuel mass we need is around 16 metric tons,

which is great because that's only 1% of the spacecraft mass.

Sounds like we're in good shape to stop this disastrous impact.

We still need to build the pulse fusion drive,

but once we have that relatively simple piece of tech,

you could successfully redirect Apophis given

that seven-year lead time.

In fact, this drive sort of makes it relatively easy.

We could make do with a less advanced option.

Now, regular rocket fuel has less than 1%

of the exhaust velocity of our pulsed future drive

and so we need almost as much initial fuel

mass as spacecraft mass.

But even that isn't completely out of the question.

I want to say congratulations to several people who

successfully answered this challenge question

and saved us from this catastrophic event.

We chose five those correct answers

at random to receive PBS Digital Studios t-shirts.

If you see your name below, send us

an email with your mailing address t-shirt size

and we'll send it out to you as soon as possible.

And be sure to join us in two weeks because we're

going to learn how to build a black hole on the next episode

of "Space Time."

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