## "Shoot for the moon. If you miss, you may hit a star."

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>-)
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Joined: Tue Apr 24, 2012 1:10 am UTC

### "Shoot for the moon. If you miss, you may hit a star."

This is not so much a logic puzzle as a fermi problem but hopefully it's ok.

The quote in title (and many, many variants) claims one who shoots for the moon may hit a star.
Find the probability of this.
In other words, compute Pr[hit a star | shoot for the moon and miss]

Given that the quote leaves many questions unanswered, below are some suggestions you may use (or not):
Spoiler:
The speed of the shot is either a: roughly the speed of the Apollo mission, or b: uniformly chosen from 0 to c
All stars will eventually go out, at which point they cease being stars and the probability drops to nearly 0
But there's also a chance an advanced civilization will deliberately act out this prophecy in order to screw with all your answers -- or reverse entropy!

And of course, I don't actually know the right answer.

taemyr
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### Re: "Shoot for the moon. If you miss, you may hit a star."

The case of the apollo mission is zero. Apollo did not have enough velocity to escape earth.

In order to hit a star other than the sun you would need enough velocity to escape the solar system. That is you would need to escape earth with an extra 42.12km/s.

In order to hit the sun you would need to cancel the orbital velocity of the earth. So you would need to escape the earth with an extra 29.72km/s.

Sandor
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### Re: "Shoot for the moon. If you miss, you may hit a star."

taemyr wrote:In order to hit a star other than the sun you would need enough velocity to escape the solar system. That is you would need to escape earth with an extra 42.12km/s.

Once you've escaped Earth, you'll be orbiting the Sun with a similar orbit to Earth's, so with an orbital velocity of around 30 km/s. It only takes another 12 km/s (in the right direction) to escape the Sun.

Anything in a circular orbit around a central body of considerably greater mass will have an escape velocity of approximately sqrt(2) times its orbital velocity. For Earth 30 km/s * 1.41 = 42 km/s. If you're in a circular orbit, it's always easier to escape what you're orbiting that it is to stop dead in your tracks and then fall into what you were orbiting.

However, is still easier to crash into the Sun than escape the Solar system. You just have to fly out past Jupiter first, in a kind of degenerate version of a bi-elliptic transfer orbit.

>-)
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### Re: "Shoot for the moon. If you miss, you may hit a star."

Well there's still a chance they'll hit a star that way

Spoiler:
They could possibly be launched into heliocentric orbit by a slingshot though -- and after that, maybe a chance encounter with a star during the Milky-way Andromeda collision in the future will result in either 1. object hitting the star directly or 2. object being launched into interstellar space.

taemyr
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Joined: Fri Jan 26, 2007 12:14 pm UTC

### Re: "Shoot for the moon. If you miss, you may hit a star."

Sandor wrote:Once you've escaped Earth, you'll be orbiting the Sun with a similar orbit to Earth's, so with an orbital velocity of around 30 km/s. It only takes another 12 km/s (in the right direction) to escape the Sun.

'True. Still outside Apollos delta-v budged though.

jewish_scientist
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### Re: "Shoot for the moon. If you miss, you may hit a star."

Lets forget about gravity, energy etc. You can just decide on a ray whose origin is your starting position, and you will move along it. You are also invincible and cannot age. Now the question is, what is the probability that a ray originating from a specific point will pass through a star. If the ray was a line of sight and the point of origin is your eyeball, then this is a restatement of Olber's Paradox. Olber's Paradox requires that at least one of these three statements be false:

1) The Universe is infinitely old
2) The Universe is infinitely big
3) stars in The Universe are evenly distributed.

Currently, cosmologists believe #1is false and the other statements are true. Because you cannot age, you have an infinite amount of time to reach your star. Whether The Universe is infinity old does not matter to you. Eventually, a star will be on your path. However, unless The Universe is static as you make your journey, then you have a problem: entropy. As time passes the entropy of The Universe increases until everything is uniform. This is called the Heat Death of The Universe. Stars will no longer exist, so traveling further along your ray will achieve nothing. There are other theories on what happens arbitrarily large amounts of time into the future, but I am too lazy to look them up.
"You are not running off with Cow-Skull Man Dracula Skeletor!"
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elasto
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### Re: "Shoot for the moon. If you miss, you may hit a star."

jewish_scientist wrote:Lets forget about gravity, energy etc. You can just decide on a ray whose origin is your starting position, and you will move along it. You are also invincible and cannot age. Now the question is, what is the probability that a ray originating from a specific point will pass through a star. If the ray was a line of sight and the point of origin is your eyeball, then this is a restatement of Olber's Paradox. Olber's Paradox requires that at least one of these three statements be false:

1) The Universe is infinitely old
2) The Universe is infinitely big
3) stars in The Universe are evenly distributed.

Currently, cosmologists believe #1is false and the other statements are true. Because you cannot age, you have an infinite amount of time to reach your star. Whether The Universe is infinity old does not matter to you. Eventually, a star will be on your path.

Also though, far enough away, space expands sufficiently quickly that stars are receding faster than the speed of light. So it doesn't help that space is infinitely big with infinitely many stars as only a finite number are reachable even if you (and they) live forever. This is also a resolution of Olber's paradox.

Feylias
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### Re: "Shoot for the moon. If you miss, you may hit a star."

jewish_scientist wrote:1) The Universe is infinitely old
2) The Universe is infinitely big
3) stars in The Universe are evenly distributed.

I'm not a scientist, and in reading the Wikipedia page for "Universe" I struggle a bit to grok even as I quibble:
Spoiler:
That said, even though the definition for "the universe" in the Wikipedia article is "everything, everywhen", that's still too vague once we start talking about the expansion of the universe and such.

To my pleb-level mind, the universe can mean one of two things:

A) All matter, and all the empty space (vacuum) where matter could exist.
B) All matter, and all empty space where matter exists or existed, but not the spaces never previously occupied by radiation or matter.

I believe that if we accept B as true, the universe has edges where matter does not now and never has existed, and beyond which nothing but pure lightless and radiationless vacuum exists. Perhaps those edges are expanding, as matter spreads itself ever farther and more thinly throughout the void, and perhaps there is no limit to how far those edges can or will expand in the fullness of eternity, but edges--and limits--they still are.

I prefer A, as a more intuitive definition. I can't conceive of edges where the empty pure radiation-less vacuum that surrounds all matter and radiation in our universe is itself surrounded by some kind of edge of its own (what would it be? Adamant? And what's beyond that?)
and so I think, minus some amazing science, infinite size and duration ought to be assumed true, and that the burden of evidence belongs with their opponents.

My real quibble is with #3, though.
Spoiler:
If we define the universe as all space that is or could be occupied by matter or radiation, there are huge swaths where that space is very likely completely empty. Maybe there are oodles of big bangs happening all the time and just spread out over the vast infinity that is ineffably distant space, but there must be emptiness between many of them, and so "evenly distributed" is problematic.

Perhaps "evenly distributed" is true on a truly vast scale (like saying that Humans are, within a certain margin of error, evenly distributed across the Earth. After all, we've got more than a few billion in both hemispheres, and this is--for certain mathy purposes, all we need to know.)

In any case, once an object escapes exceeds the Moon's distance from the Earth without impacting it, it may end up:
Spoiler:
Falling back into the Earth.
Falling into orbit around the Earth or the Moon.
Impacting or falling into orbit around another large non-star mass (like Jupiter, a ring of Saturn, or a wayward asteroid that already has the mass to absorb the inertia and an orbit of its own.)
Falling into a black hole (though this may count as a star for some purposes; it likely contains things that were stars previously, I think.)
Orbiting a star without ever impacting it.
Somehow avoiding all the possible gravity wells for a very long time despite being attracted to them, and infinitely wandering ever toward the horizon of the universe (though, given its sub-light-speed travel, the edge of the occupied universe will ever be expanding faster than this ill-fated star-seeking projectile of ours.)
Spontaneously turning into some flowers and a large cetacean (deep science.)

Of course, it could also end up pulled into a star (or being knocked into one by an impact with another object.)

I intuit (bad science!) that while stars are very big and attractive, getting the object to avoid planets, planetoids, asteroids, comets, and an orbit within the solar system (unusually replete with planets and concomitant junk, I think) is very difficult. Our best bet of getting this thing into a star is to aim at Sol, but aiming at a star is cheating. Instead, to fit with the spirit of accident, we'd best just aim to miss anything within the solar system and ensure we have enough speed to avoid all of the above. (Hard to do when you were merely aiming at the moon.)

Spoiler:
If we were sincerely aiming at the moon and missed, the odds of falling into any star with such small velocity are negligible because we're more likely to end up in orbit around something or impacting something else. (Negligible is "incalculably small", at least by virtue of any math I could figure out in this twenty-four-hour period)
At least, within a few billion years. After all, even if you end up firmly embedded in Venus, eventually there's a real chance that Venus itself will end up merged with one or more stars as Sol ages, possibly explodes, or--for all I know--the galaxies stop flying apart and start falling back together. I think there's a very real chance that every one of us will eventually end up merged with the material that currently comprises Sol (though our great grandprogeny will have long since died by then.) I'm not the right kind of scientist to know whether this is more likely than the stars simply winking out and ceasing to be "stars" without causing enough chaos to suck planetoids into their mass.

I think the spirit of the question could be reworded as: If we could shoot a thing at the moon and contrive to miss, and this thing had the ability to henceforth ignore gravity and simply travel in a straight line until it hit something, how likely would it be to hit a star (rather than a non-star or nothing at all.)

If falling into an orbit around a distant star counts as "hitting a star" for our purposes, well, that puts our present state of orbit around Sol in a new light. High five!

Also, Frungy is awesome.

CharlieP
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Location: Nottingham, UK

### Re: "Shoot for the moon. If you miss, you may hit a star."

Feylias wrote:I think the spirit of the question could be reworded as: If we could shoot a thing at the moon and contrive to miss, and this thing had the ability to henceforth ignore gravity and simply travel in a straight line until it hit something

According to General Relativity, things affected by gravity are travelling in a straight line.
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