Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Mariana on 15/02/2019 09:50:44
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Charles wants to know:
Can a scientist please help me calculate an asteroid's impact time? I'm writing a sci-fi story.
The asteroid is about 2.5 astronomical units from the sun in the asteroid belt. The protagonist wants to strap rockets to the rock and send it rocketing to earth, in orbit. I was told that if the asteroid was 90 degrees ahead of the earth, it could be launched at a velocity of 23 km/s-1 directly at the sun, and will arrive 3 months later. Is this correct?
Any other details (velocity at arrival at earth, etc) would be appreciated.
Can you help?
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To get an asteroid to Earth, you need to reduce or stop its orbital speed. From there, the sun's gravity will take it to Earth.
For a simple scenario (dropping straight in), you remove the 25 km/sec speed of the asteroid and it drops to Earth, arriving at about 12.3 km/sec but then Earth's gravity adds some more speed so it comes in at over 14 km/sec as it nears Earth. You'd need to impart 30 km/sec to the thing to match Earth's velocity plus take off all/most of that 14 km/sec and possibly leave some speed to keep it in Earth's orbit. I compute at least 33 km/sec of change in velocity to get it in Earth orbit. That's a lot of effort.
How long to drop in? It has 1.5 AU to travel at around 5 km/sec average which takes something near 1.3 years to arrive. Your idea has it not starting from a halt but actually having an initial speed of 23 km/sec straight in (in addition to the effort needed to stop its forward motion). At that speed, I get around 100+ days, which is pretty close to your 3 month estimate. My calculation was fairly crude.
It is implausible that any engineer would use such inefficient methods. For the story to be believable, they'd want to minimize energy requirements, so all you'd do is trim off some of the asteroid's orbital speed, putting its orbit into an ellipse with perihelion at Earth. Once it gets near Earth, a second braking thrust is needed to bring its aphelion down to match us. Then the only additional braking action needed would be that required to bring it into Earth orbit, and that can get a good bit of help from a slingshot around the leading side of the moon. The process done at modest power takes years, but that is far preferable to the brute method of having it come straight in and then having to suddenly execute another 90° turn to match Earth's motion. Doing it with two hard/short burns as described above might requires about 1.5 years between the two burns.
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The protagonist wants to strap rockets to the rock and send it rocketing to earth
Is this a protagonist (the good guy) or the evil villain?
- Presumably, the evil villain would want it to run into Earth at the maximum possible speed, so he wouldn't need the braking burn that Halc talked about.
calculate an asteroid's impact time?
What orbit you take depends on how quickly you want to get to your destination, and how much fuel you can afford to burn (faster = burn a lot more fuel!).
- The Japanese space probe Hayabusa2 is currently at the asteroid Ryugu, and later in February 2019 plans to collect a small sample of this asteroid to return to Earth (1 - 30 grams).
- It took 4 years to reach the asteroid, and will take 1 year to get back to Earth
- The H-IIA 202 launcher has a mass of around 280 tons at launch, most of which is fuel.
- So you are looking at over 10 tons of fuel launched from Earth to return 1 gram of asteroid safely to Earth.
- This is despite the space probe using the very efficient ion propulsion drive (which is also very slow)
See: https://en.wikipedia.org/wiki/Hayabusa2
This mission was also mentioned on the 11 Feb 2019 Space Boffins podcast (https://www.thenakedscientists.com/podcasts/astronomy-podcasts/space-boffins)
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You're driving to hard a bargain Charles. Do you believe in 'gravity'?
You shouldn't concern yourself with the numerics. You just need a logic that works for the story.
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If our antagonist, where to completely stop the asteroid in its orbit and just let it fall in towards the Sun, It would take 223.5486051 days to do so. More than what giving it a inward push would be, but less than what an minimum energy trajectory would be. This would require a delta v of 18.84 km/s, compared to the 7.27 km/s for a minimum energy trajectory. If we are limited to chemical rockets, This would require ~4 kg of fuel per asteroid for the low delta v trajectory and ~65 kg of fuel per kg for the high delta v one.*
There are rocket engines that are more efficient, such as ion engines, but there are drawbacks. Ion engines are low thrust, so you aren't going to just be able to change the asteroid's trajectory in one go, but as a slow constant burn. The asteroid would spiral in over a long time. Ion engines use materials for the reaction mass that aren't as common as the hydrogen and oxygen used in a chemical rocket. The probe mentioned by evan_au uses xenon, a noble gas that makes up ~ 1 part in 20 million of our atmosphere. So even though a ion engine would lower the total fuel considerably, You'd still need to somehow produce a lot of xenon (the probe used just 66 kg.)
Other considerations to take: Above I gave a time for a direct fall out to several decimal places. In this instance, timing will be critical. The Earth orbits at ~30 km/sec. Which means it moves the width of its diameter every 7 min. The asteroid arriving just 10 min too late or too early could result in a miss (you'd have to work out the impact parameter which takes the Earth's gravitational effect on the asteroid to get an exact answer) The Earth's orbital velocity and orbital distance from the Sun also vary over the course of its orbit, so these also play a factor in getting the timing correct.
Another factor is that, unless our scientist is very lucky, the asteroid is not likely to have the same inclination of its orbit as the Earth. Their orbits will not be in exactly the same plane. So even if you drop your asteroid straight in, it would very likely miss the Earth by passing either "above" or "below" its orbit. If you could "drop" the asteroid when it was at one of the 2 points of its orbit where the two planes cross (called nodes), and the timing was such that it arrived at the Earth at the same time that the Earth was crossing one of its nodes, then your could avoid this, But this would mean waiting until just the right alignment between asteroid and planet occurs, assuming it ever does.
The only other resort is to adjust the trajectory to compensate. (with the low delta v trajectory this is done with the "broken plane maneuver", with a burn done when the trajectory crosses a node.)
Have to sign off, as my computer just warned me that it is about to restart for an update.
*This doesn't factor in what would be needed if you had to carry the fuel to the asteroid to start with.
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Damn, I enjoyed reading you Janus :)
there are some pretty nice people here.
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I suggest that your protagonist is able to beg, borrow or buy a hydrogen fusion drive, which should increase the amount of asteroid he can return by a factor of 100,000 or so.
Even better, if (s)he can use the solar wind as fuel, that improves the ratio even more.
Even better, if it is already in Earth orbit, that reduces the initial fuel load considerably!
See: https://en.wikipedia.org/wiki/Bussard_ramjet