100 Years of Relativity Challenge Winners Space Time PBS Digital Studios


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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. [THEME MUSIC] 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." [THEME MUSIC]