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Today we have a solution to our killer asteroid challenge
episode, but before we get to that
there's something very special about today.
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."