# Help with equations for deflecting an asteroid

This is a homework assignment, but not the physics part. I have to write a python script to simulate the size of a warhead required to deflect an asteroid (parameters inputted by user) that is on a collision course with earth. What is constant is that the missiles velocity is 11 km/s, the largest warhead created is rated at 50000 megatons, and the save distance from earth is 3 time the radius of earth from earth's center. We are also assuming half of the energy of the missile will be used to propel the asteroid while the other half will be used to deflect it.

How do I calculate the minimum distance from earth required to deflect it enough from earth to pass 3x earths radius? I also need to make some calculations with smaller missiles, but I figure once I know the needed equations I can figure this out.

I know how to program but it has been some time since I took physics and I went through my old book, test, and homework and cant seem to find anything about explosions that will help me.

Thanks in advance.

EDIT:

Here is a diagram given to help illustrate the problem I am to write the code for.

The inputs for the program are:

• the asteroids velocity,
• diameter (assume its a ball),
• type ( from type we estimate density and from there mass).

The outputs for the program are:

• the energy of the missile used for the calculation with the max being a 50000 megaton
• distance to impact
• missiles time of flight
• lead time required

I am pretty sure I can get time of flight and lead time and use different energies but the distance is what is throwing me off.

• what I should have said is what is known or assumed not what is constant. – Jeffery Baugh Sep 27 '14 at 18:35
• Can you write a Python workaround (like a lot of frameworks or applications), and use sunlight and huge mirrors to deflect the asteroid? It would be cheaper, and you wouldn't have to build a warhead and wait for it to get there. :-) – MacGyver Sep 27 '14 at 19:19
• Between this question, this now-deleted question, and this question on another site I really have to wonder who is assigning all these asteroid deflection questions. – user10851 Sep 27 '14 at 19:20
• @ChrisWhite - If a masters thesis (and occasionally a PhD thesis) in aerospace engineering counts as "homework", asteroid deflection was assigned as a rather common "homework" problem, fifteen years ago or so. – David Hammen Sep 28 '14 at 16:52
• @MacGyver unfortunately I can't do a work around. Though I feel the assignment is not extremely detailed or leaves a lot to assumption I am to used the diagram and method of deflection for my program. – Jeffery Baugh Sep 28 '14 at 18:15

## 1 Answer

A "collision course" is a very fuzzy concept: if you are "barely going to hit" you are on a collision course but don't need a lot of deflection. However, let's assume for a moment a stationary earth, a meteorite of mass $m$ at distance $D$, heading for earth of radius $R$ with velocity $v$.

The equations you need are conservation of angular momentum and energy. If you give the asteroid a lateral kick $F\Delta t$ (impulse = change in momentum $m\Delta v$), then the angular momentum of the new orbit is

$$L = m \Delta v \cdot r$$

And this will still be the angular momentum when you reach earth. So the question then becomes, what velocity will it have when it gets to earth? The answer - at a distance of $3R$ it will have increased its velocity (because of the gravitational potential energy it had). Assuming you started very far away ($r>>R$) you find this from

$$\Delta E = \frac{GmM_e}{3R} = \frac{gmR}{3}$$

since we know that the gravitational acceleration of the earth at the surface, $g$, is given by

$$g = \frac{GM_e}{R^2}$$

Finally, kinetic energy is $\frac12 m v^2$ as you may remember. So velocity will increase when energy increases.

I'm going to leave you the fun of combining these equations to solve your problem. I must admit I am a little bit unsure of how you plan to model the explosion - you might want to elaborate a bit on that if you need help figuring out the impulse $F\Delta t$.

• Thank you for your comment. It looks like this may help me some but I don't know. I added a diagram and input/outputs of my program to help illustrate the problem to be programmed. – Jeffery Baugh Sep 28 '14 at 18:14
• Ok can you help me with one more thing, and maybe I am missing this, but how do I calculate the minimum distance (D) that missile collision needs to take place to miss us at the 3R distance? – Jeffery Baugh Sep 28 '14 at 22:08
• If you know the impulse you can deliver, the angular momentum equation will tell you how far away you need to be. I cannot write it out on my phone - May get near a computer in a few hours if you are still stuck. Note my $r$ is your $D$ – Floris Sep 28 '14 at 22:13
• Ah that actually helps a lot. I was viewing r and D two separate values. Thanks again. I will be finishing this program up tomorrow and I will let you know if I get stuck. I think I can calculate the impulse as it was in my physics book. – Jeffery Baugh Sep 28 '14 at 22:43