The Physics of Smashing a Spacecraft Into an Asteroid

Two weeks ago, NASA hit a small asteroid called Dimorphos with a spaceship. The Double Asteroid Redirection Test is also known as DART. If you missed it, here is a video taken before the crash.

NASA did this because they wanted to see if a collision could divert an asteroid. This was just a test. Dimorphos does not have a trajectory that would pose a threat to us. We don't know how they did it.

Let's dig into some of the most interesting physics aspects of the mission while we wait for the space agency to release their calculations.

The size and speed of the particles.

I'm going to start with a video analysis. Is it possible to see the video and see a plot of the position of the craft? Absolutely! This is how it happens. The DART has two cameras, one for Didymos and another for Asteroids. The camera has an angle of view of 0.29 degrees. You would see an angle of 0.29 degrees if you drew a line from the left side of the camera to the right side.

It looks larger when you get closer to it. A person stands at the other end of a parking lot. Hold out your thumb by stretching out your arm. It is1-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-6556 is1-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-6556 Because a thumb isn't actually bigger than a human, the object'sangular size is what you're seeing.

There is a relationship between the real size of the object and the radians of the object.

The apparent size of Dimorphos is shown in the video, and the actual size is shown here at 170 meters. But what about the times? The NASA video runs at 25 frames per second, but it isn't real time. It is increased by 10. The time between the two frames is less than one second.

I need to get some frames from the impact video, measure the Dimorphos' size, and use that to calculate the distance. If I have a time plot and a position plot, I can see how fast the ship is moving. The relative speed of DART with respect to the asteroid will be determined by the change in position with respect to time. The plot is here.

This isn't the best way to measure the speed of the craft. You can see that I fit some linear functions to the data. The green line is the best place to put the data. The slope of the line gives a fast speed.

The impact speed was 22,530 kilometers per hour. The red line is the final part of the data. This second fit has a slope of 7.7 km/s. The method still gives a rough idea of the final speed before the crash.

There are elastic and inelastic crashes.

When NASA releases its analysis, we can learn how much DART affected the asteroid, as well as how much damage was done to it. Let's look at some of the aspects of the collision that are being studied.

DART had an impact on the asteroid. Since forces are always exerted between objects, the asteroid also exerts a force with the same magnitude on the spaceship. If there are no other forces on the system, then a conserve of momentum will occur.

It is defined as the product of an object's mass and its speed.

The total momentum after the collision must be the same as the total momentum before the collision. There must be no external forces on the system in order for this to happen.

There are a variety of types of crashes. An inelastic collision is when two objects stick together, like a piece of clay hitting a basketball. Their final velocities must be the same after the crash.

The elastic collision is at the other end of the spectrum. Think of two rubber balls flying apart after they collide. Both momentum and energy are used in a collision. The energy of an object can be defined as follows.

After the collision, the sum of energy for two objects has the same value as before.

This has to do with a spaceship colliding with a rock. It matters how elastic you are. I'm going to show you an example of a collision between a spaceship with a mass m D and an asteroid with a mass m a that starts off at rest. The spaceship sticks into the asteroid after the crash. The final speed of the objects will be v 2.

The final velocities of the two objects can be solved using the initial and final momentums of the two objects.

There are some numerical values from the DART impact. The spaceship has a mass of 610 kilograms and is travelling at a 6 kilometer per second speed. Dimorphos has a mass of about 10 kilo. The final speed is 0.73 millimeters per second. That's correct, it's small.

I thought the asteroid began with zero speed. For a moving target, 0.73mm/s would be the change in speed.

Let's say that the spaceship has a collision with the rock. It won't stick to the asteroid, but it will bounce off and save the system's energy. The "D" and "a" subscripts need to be included in the velocities of both Dimorphos and DART.

The two equations I get now are related to the conserved of energy.

There are a few things to notice DART moved backwards after the collision because it bounced. This one-dimensional example will have a negative momentum due to the fact that velocity is a vectors.

The square of the velocity is dealt with in the equation. DART has positive energy even though it has a negative speed.

We have two equations and two variables that are easy to solve. If you did the math, you'll see what I'm talking about. I have all the information you need.

The final velocities of DART and Dimorphos are used. The recoil velocity is twice what it is for the collision. The DART spacecraft has a bigger change in momentum since it bounced back. This means that Dimorphos will have a bigger change in speed. It's still a small change, but twice it's bigger than the other way around.

Two extremes of the collision spectrum are elastic and inelastic. The objects don't stick together but the energy is not conserved. The best way to change the trajectory of an asteroid is by hitting it with something.

It looks like there is at least some material ejected from the asteroid after the collision. Since the debris moves in the opposite direction of DART's original motion, it seems that the spacecraft partially bounced back. If you want to budge a space rock, you want to see what that looks like. If no ejected material was present, you would have something close to an inelastic collision.

The result of the impact can be measured.

The best-case scenario would change the asteroid's speed by just 1.34 millimeter per second. It's difficult to measure a velocity change of this size. Part of a double asteroid system, Dimorphos has a bonus feature. It is around Didymos. NASA chose this target due to that. If you want to find out the effect of a crash into Dimorphos, you need to know how long it takes for the object to make a complete circle, and how long it takes for it to burn up.

The moon is made to circle the Earth by the same physics as dimorphos. Dimorphos are pulled towards their common center of mass by Didymos because it is larger. The two objects will eventually collide if they both start from rest. That is not true. Dimorphos moves in an ellipse around the center of mass because it has avelocity that's mostly parallel to the force of gravity. It's1-65561-65561-6556 is1-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-6556

Didymos is pulled on by Dimorphos, which is at the center of mass. The Didymos, the larger of the two asteroids, has a small, almost unnoticeable, circle around the center of mass.

Both asteroids have the same period of time. You can determine their mass by knowing what it is and how far away they are. There is a trick. The sum of their mass is only given by the orbital period. Dimorphos and Didymos would have the same density if you assumed they were made of the same material. It is possible to determine both mass using that and relative sizes.

The code for the Python model of the asteroids can be found here. This isn't live. People don't want to watch an animation that long.

It's time for the fun part. If DART collides with Dimorphos, it could have a big change in speed. What do you think would happen to the motion in the air? Let's see how the model works.

This is an animation. Didymos and Dimorphos are before the collision. There are two sets of asteroids in yellow that show the motion after the impact of the spaceship.

There are some intriguing things to notice. The model shows that after the DART crash, the Dimorphos' trajectory changed. Dimorphos slowed down and moved into a slightly non-circular circle. But what about the last part of the year? The yellow version of the asteroid completes an elliptical path before the undisturbed one. This is exactly what would happen if the Orbital Motion was not intuitive.

It is possible to see the difference in periods with a graph. The plot shows the horizontal position of Dimorphos. The red curve shows the movement after the impact with DART.

It's not easy to see the exact motion of an asteroid. It is too small and close to the Didymos. We have a way to measure the time it takes for a single full moon.

You could see the light reflected from the asteroids. Light intensity can be detected by a telescope on Earth. You can't see it from Earth if the smaller asteroid is behind the larger one. When it is behind the bigger one, the intensity of light will decrease, but it will increase again when it comes back. The change in intensity of light can be used to calculate the orbital period. It was the result of the DART impact if it has changed. It's cool.

The question is still unanswered, would this bumped spaceship make enough of a difference to prevent an asteroid from hitting the Earth? It depends as is often the case. If the asteroid is on its way to Earth, it won't make a difference. If you can impact an asteroid when it's still far away, even a small change in speed can be enough to cause a collision with our planet. It's what we want, but we need to know what happens when a spaceship collides with an asteroid. The DART mission is about that.