Asteroid deflected by Earth -- effect on Earth

In summary, the conversation discusses finding the proportional change in the kinetic energy of the Earth and the change in the semi-major axis of its orbit caused by an approaching asteroid with a mass of 6 \times 10^{20} \textrm{kg}. It also mentions finding the distance of closest approach and angle of deflection for the asteroid in the frame of reference where the Earth is at rest. The conversation includes equations and attempts at solving the problem, but it is unclear if the approach is correct.
  • #1
Dazed&Confused
191
3

Homework Statement


Suppose the asteroid of [other problem] has a mass of [itex]6 \times 10^{20} \textrm{kg} [/itex]. Find the proportional change in the kinetic energy of the Earth in this encounter. What is the change in the semi-major axis of the Earth's orbit? By how much is its orbital period lengthened?

Other problem:
Suppose the asteroid of [other problems] approaches the Earth with an impact parameter of [itex]5 R_E[/itex], where [itex]R_E[/itex] = Earth's radius, moving in the same plane and overtaking it. Find the distance of closest approach and the angle which the asteroid is scattered, in the frame of reference in which the Earth is at rest.

Homework Equations

The Attempt at a Solution


[/B]
The value of Earth's velocity is 28.9km/s and that of the comet is 38.6km/s. The angle of deflection was 18.4 degrees. I tried calculating
[tex] \dot{\textrm{R}} - \frac{m_1}{M} \dot{\textrm{r}} [/tex] to work out the new velocity and from that the energy change. The book's answer was [itex]3\times10^{-6}[/itex] whereas mine was [itex]59\times 10^{-6} [/itex]. I'm not sure if my approach is correct.
 
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  • #2
The angle of deflection is not sufficient, you also have to know the initial or final direction or something else about the geometry of the problem.

Where does your formula come from?
 
  • #3
Dazed&Confused said:
I tried calculating ##
\dot{\textrm{R}} - \frac{m_1}{M} \dot{\textrm{r}}## to work out the new velocity and from that the energy change.
It is not clear what your reasoning or algebra is from that decsription. Please post the details of what you did and on what basis.
 
  • #4
OK I uploaded a couple of images. The first shows the set up and the second what has happened in the reference frame of the Earth.

Here [itex]\dot{\textrm{R}}[/itex] is
[tex]
\frac{m_1 \dot{\textrm{r}}_1 + m_2 \dot{\textrm{r}}_2}{m_1+m_2}
[/tex]
which is the velocity of the centre of mass and [itex]\dot{\textrm{r}} = \dot{\textrm{r}}_1-\dot{\textrm{r}}_2[/itex]. The formula I quoted is that of [itex]\dot{\textrm{r}}_2[/itex]. Here [itex]\dot{\textrm{r}}[/itex] is the velocity of the comet in Earth's reference frame. I find Earth's velocity components with this formula and then this speed to find the kinetic energy.

Edit: for some reason the text did not show up, but here Earth is traveling at 29.8km/s, the comet at 38.6km/s, and the angle is 18.4 degrees.
 

Attachments

  • cometearth1.pdf
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  • cometearth2.pdf
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  • #5
The relative velocity between Earth and asteroid does not change, and conservation of momentum gives the final velocity of all components. Start there, then you can be sure to have the right formulas.
 
  • #6
What about the equation [itex] \mu \ddot{\textrm{r}} = \textrm{F}[/itex]? Wouldn't [itex]\dot{\textrm{r}}[/itex] change in this case? Also the deflection angle is the one you calculate from the previous question, which is confirmed in the solutions.
 
  • #7
Also the formula for [itex]\dot{\textrm{r}}_1[/itex] is [itex] \dot{\textrm{R}} + m_2/M \dot{\textrm{r}} [/itex] and we find that the total momentum is [itex] m_1 \dot{\textrm{r}}_1 + m_2 \dot{\textrm{r}}_2 = M \dot{\textrm{R}}[/itex] so that momentum is conserved according to them.
 
  • #8
Do you know F as function of time? If not, it doesn't help.

What are r1, r2? You are wildly combining various different formulas and symbols without introducing them or showing where they come from. That does not help.
 
  • #9
The previous question was from chapter 4 of the book and had a section on orbits from inverse square law forces. It had subsections on hyperbolic orbits and the deflection, from which I was able to obtain the previous result.
 
  • #10
Sorry I didn't notice your question. Here [itex]\textrm{r}_1[/itex] and [itex]\textrm{r}_2[/itex] are the positions of the comet and Earth, respectively.
 

Related to Asteroid deflected by Earth -- effect on Earth

1. How will the asteroid deflected by Earth affect our planet?

The effect of an asteroid deflected by Earth depends on various factors such as the size, speed, and composition of the asteroid. If the asteroid is small and made of rock, it may disintegrate upon entering Earth's atmosphere. However, if the asteroid is large and made mostly of metal, it may cause significant damage upon impact.

2. Can an asteroid deflected by Earth cause a tsunami?

Potentially, yes. If the asteroid hits the ocean, it can displace a large amount of water, resulting in a tsunami. However, the chances of this happening are low, as most asteroids that come close to Earth are relatively small and would not cause a significant tsunami.

3. How does Earth's atmosphere affect the deflection of an asteroid?

Earth's atmosphere plays a crucial role in deflecting asteroids. As the asteroid enters Earth's atmosphere, it experiences friction and heating, causing it to change its trajectory. This effect, known as the Yarkovsky effect, can either increase or decrease the asteroid's deflection, depending on its size, shape, and composition.

4. Will the deflection of an asteroid by Earth affect its future orbit?

Yes, the deflection of an asteroid by Earth can alter its future orbit around the Sun. If the asteroid's trajectory is significantly changed by Earth's gravity, it may end up in a different orbit or even get ejected from the solar system. This effect is known as the gravitational slingshot effect.

5. What measures are being taken to prevent an asteroid impact on Earth?

Scientists and space agencies around the world are constantly monitoring near-Earth objects and developing methods to prevent an asteroid impact. These include asteroid deflection techniques such as using a spacecraft to nudge the asteroid off its course or using a gravity tractor to alter its trajectory. However, these methods are still in the research and testing phase and may not be effective for larger asteroids.

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