Hypothetical question about objects in the asteroid belt

"Arie Kazachin" om wrote
in message ...

Suppose that all (most) of the objects in the asteroid belt had been
brought
together till they're touching one another and then left alone.
Will their combined gravity hold them together or the Jupiter's
tidal forces will spread them again to form a belt?

Is there a "critical mass" above which they'll remain together and below
which they'll not?

I'd expect they'd stay stuck together. Jupiter's influence kept tiny
fragments from combining into a planet, rather than disrupting an existing
planet, which is what you'd practically have by that point.

Still, it'd be a pretty dinky planet; smaller than our moon.

There are much better uses for the belt than making another worthless
planet...

--


Regards,
Mike Combs
----------------------------------------------------------------------
We should ask, critically and with appeal to the numbers, whether the
best site for a growing advancing industrial society is Earth, the
Moon, Mars, some other planet, or somewhere else entirely.
Surprisingly, the answer will be inescapable - the best site is
"somewhere else entirely."

Gerard O'Neill - "The High Frontier"

(Arie Kazachin) writes:

Suppose that all (most) of the objects in the asteroid belt had been brought
together till they're touching one another and then left alone.
Will their combined gravity hold them together or the Jupiter's
tidal forces will spread them again to form a belt?

No. The reason why it is commonly said that Jupiter "prevented" a planet
from forming in the Asteroid Belt is that the gravitational perturbations
Jupiter induced in asteroidal orbits caused objects to be ejected from the
belt faster than they could clump together. However, since the asteroids
are WELL outside Roche's limit for Jupiter, they would indeed be able to
"stick together" if they collided, =ASSUMING= you could bring them together
_gently_ enough --- and therein lies the problem: It is not possible to
bring two bodies together at a velocity smaller than the escape velocity
for the combined body, so when two large asteroids collide, they are more
likely to smash each other to bits that escape than to fuse together.
To accrete mass, one of the two bodies much be much larger than the other.


Is there a "critical mass" above which they'll remain together and below
which they'll not?

No. Roche's limit does not directly depend on the mass of the smaller object
(except that it must be much much smaller than the primary). What Roche's Limit
depends on is the ratio of the _densities_ of the two objects, and the distance
between them in units of the primary's radius. For a given primary density
and a given number of planetary radii from the primary, there is a critical
_density_ for the smaller object, below which it will not be able to hold
itself together. The asteroid belt is so many jupiter-radii away from Jupiter
that virtually any solid or liquid substance would be able to hold itself
together against Jupiter's tidal forces.


For a layperson's summary of Roche's limit, see:

http://www.world-builders.org/lessons/less/les1/moons/roche.html

For a simple heuristic derivation of Roche's limit, see:

http://www.merlyn.demon.co.uk/gravity3.htm#Roche.


-- Gordon D. Pusch

perl -e '$_ = \n"; s/NO\.//; s/SPAM\.//; print;'

In message -
(Gordon D. Pusch)23 Mar 2004 13:02:59
-0600 writes:



For a layperson's summary of Roche's limit, see:

http://www.world-builders.org/lessons/less/les1/moons/roche.html

For a simple heuristic derivation of Roche's limit, see:

http://www.merlyn.demon.co.uk/gravity3.htm#Roche.


-- Gordon D. Pusch

Thanks!

************************************************** ****************************
* Arie Kazachin, Israel, e-mail: *
************************************************** ****************************
NOTE: before replying, leave only letters in my domain-name. Sorry, SPAM trap.

In message - "Mike Combs"
Tu e, 23 Mar 2004 12:58:20
-0600 writes:



Still, it'd be a pretty dinky planet; smaller than our moon.

There are much better uses for the belt than making another worthless
planet...


Except maybe making somewhat more predictable the trajectories of NEOs
over LONG periods of time...


************************************************** ****************************
* Arie Kazachin, Israel, e-mail: *
************************************************** ****************************
NOTE: before replying, leave only letters in my domain-name. Sorry, SPAM trap.

Arie Kazachin wrote:

In message -
(Gordon D. Pusch)23 Mar 2004 13:02:59
-0600 writes:


For a layperson's summary of Roche's limit, see:

http://www.world-builders.org/lessons/less/les1/moons/roche.html

For a simple heuristic derivation of Roche's limit, see:

http://www.merlyn.demon.co.uk/gravity3.htm#Roche.


-- Gordon D. Pusch

Thanks!

************************************************** ****************************
* Arie Kazachin, Israel, e-mail: *
************************************************** ****************************
NOTE: before replying, leave only letters in my domain-name. Sorry, SPAM trap.



Ok, 550 thousand CD's in a slightly tilted, very circular Lunar orbit?

2.423 * Radius of Luna * density of Luna
all divided by density of a CD

So...

2.423 * 1 (Radius of Luna) * 3.34 (density of Luna)
all divided by the density of a CD disk
(greater than 1, cuz they sink in water)
(anybody???)

(Arie Kazachin) wrote in message ...
In message - "Mike Combs"
Tu e, 23 Mar 2004 12:58:20
-0600 writes:



Still, it'd be a pretty dinky planet; smaller than our moon.

There are much better uses for the belt than making another worthless
planet...


Except maybe making somewhat more predictable the trajectories of NEOs
over LONG periods of time...


************************************************** ****************************

* Arie Kazachin, Israel, e-mail: *
************************************************** ****************************
NOTE: before replying, leave only letters in my domain-name. Sorry, SPAM trap.


The best way to predict something is to control it certainly. We do
have the capacity to find and change orbits of objects in space.
Defense against an errant asteroid is an interesting benefit. But,
more profitable is the ability to find and capture very rich asteroids
and bring them into useful orbits around Earth.

Once in Earth orbit, we can then place remotely controlled factories
that use solar power to process the asteroid into useful products.
These products may be used in space, to expand our capacity there, or
sent back to Earth, to pay the bills.

Ultimately, we'd be able to transfer all our productive capacity,
including farms and forests, to space. This would leave the Earth as
a vast natural preserve, in which humans lived. People would report
to work via remote control through a wireless broadband connection
supported from space.

Products would descend like JDAMs, and come to a soft landing near
each customer. Delivery would occur in minutes. Costs would be very
low.

How much energy would it take to move an asteroid into Earth orbit?


http://nssdc.gsfc.nasa.gov/planetary...eroidfact.html

Well, let's look at Ida;
NAME SIZE MASS ROTATION PERIOD TYPE AXIS ECCENT. INCL.
243 Ida 58 x 23 100 4.633 hrs 4.84 yrs S 2.861 AU 0.0451 1.14 deg

So Ida is in a nearly circular orbit 2.861 times the distance from the
Sun Earth is. Its 58x23 km and masses 100x10^15 kg. Its an 'S' type
asteroid, so likely not of much interest to humanity for that reason.
But, Ida is typical of the orbit and mass of the asteroids any company
that made money moving industrial asteroids would be asked to tackle.
Those responsible for nudging dangerous asteroids into less dangerous
orbits have a far easier task.

Since the masses are so large, minimum energy is an interesting
approach. This calls for Hohmann transfer orbits.

http://liftoff.msfc.nasa.gov/academy...s/hohmann.html
http://scienceworld.wolfram.com/phys...sferOrbit.html

To simplify things, we'll assume no plane change, this gets us a rough
estimate of the total energy needed with Ida since it has a low
inclination. ANd lets us use simpler math since we don't have to
include 3D vector analysis which plane changes require.

So, the circular orbit Ida is now in is 2.861 AU
When Ida is captured in Earth orbit, it willbe in an orbit with 1.000
AU

The minimum energy transfer orbit will have 1.931 AU semimajor axis,
with an apoapsis of 2.861 and periapsis of 1.000

The velocity of Ida now is 17.635 km/sec taking the circumference of
the orbit and dividing it by the period.

The velocity of Earth now is 29.865 km/sec by the same procedure.

This implies a constant [mu] of 890 given the units and so forth - so
for our transfer orbit we have;

at apoapsis

v = SQRT(890*(2/r - 2/(r1+r2)))
= SQRT(890*(2/2.861 - 2/3.861))
= SQRT(890*(0.6991 - 0.5180))
= 12.694 km/sec

and at periapsis

v = SQRT(890*(2/r - 2/(r1+r2)))
= SQRT(890*(2/1 - 2/3.861))
= SQRT(890*(2 - 0.5180))
= 36.317 km/sec

Now, a circular orbit at the apoapsis as we've already seen is 17.635
km/sec so the difference in speeds are;

17.635 - 12.694 = 4.941 km/sec

which is the requirement of the first impulse

And, a circular orbit at the periapsis as we've already computed is
29.865 km/sec so the difference in speeds are;

36.317 - 29.865 = 6.452 km/sec

And we're slowing down both times, which is consistent with lowering
our altitude from 2.861 AU to 1.000 AU

We need a total delta vee of

4.941 + 6.452 = 11.393 km/sec

To minimize energy usage in our rocket, its best to make the exhaust
speed equal to the delta vee requirement. This implies an exhaust
speed of 11.393 km/sec.

This is easily achieved by nuclear pulse rocket technology. We'll
vaporize part of the asteroid itself to produce exhaust - so the
asteroid itself becomes the propellant.

This is achieved by building small atom bombs that direct their energy
toward the asteroid and vaporize a small controlled portion of it.

In any case, we can figure the amount of the asteroid we've got to
throw away - this is given by;

u =1- 1/exp(1) = 0.6321

which is a lot, but if we want to minimize energy usage, that's what's
called for.

Of course, we could increase exhaust speeds to limit wastage, but this
wastes energy. For example, if we have twice the exhaust speed as
delta vee required, then we throw away;

u = 1 - 1/exp(1/2) = 0.3935

Then, we could go to 3x required speeds in our exhast velocity to
obtain the following propellant fraction;

u = 1 - 1/exp(1/3) = 0.2835

Clearly the relative costs of energy versus the material you're moving
will determine what exhaust speed a company that makes money moving
material will choose.

Knowing the starting mass, exhaust speed, and propellant fraction we
can now compute the total energy required in the exhaust jet;

E = 1/2 * m * V^2
= 0.5 * 100e15 kg * 0.3935 * (2*11,393 m/s)^2
= 10.215e24 joules

That's 48.6 million times a 50 megaton blast - the largest explosion
ever achieved by humanity.

Now, no rocket system is perfectly efficient. There are always
losses. But, most systems, even nuclear pulse systems, especially
when moving large masses, are better than 0.5 - so, we would likely
need 20e24 joules of energy applied as indicated in two pulses, to
move this asteroid.

This is about 100 million times a 50 megaton blast.

http://nuclearweaponarchive.org/Library/Teller.html

Now, something 10,000x as explosive can be built by adding two stages
to a teller ulam bomb - which reduces the number of bombs to only a
10,000 blasts 500 Gigatons each.

10,000 is a nice round number. It gives you a lot of control,
imparting only 1 meter per second per blast, and its a reasonable
number to work with,

How big would each bomb be physically before detonation?

http://nuclearweaponarchive.org/Nwfaq/Nfaq12.html

Well, if we used Li6/D - we'd get 64.0 kilotons yeild per kg mass.
So, a 500 gigaton bomb would mass around 8,000 tons.

Ten thousand of these puppies would mass 80 million tons.

The size of the largest tanker ever built is around 500,000 tons.

So, we're talking massive spacecraft here.

But, the largest spacecraft ever conceived is in this size range. A
nuclear pulse spaceship designed for interstellar travel was worked on
briefly during project Orion that at their largest would mass nearly
80 million tons.

It really takes very little fissile material to set off massive
quantities of fusion meterials. We can use the same technique
developed for inertial confinement fusion to compress tiny amounts of
fissile materials - to use as a spark plug. As little as a few grams
of plutonium perhaps per 8,000 ton bomb.

We can imagine then a fleet of 300 or so massive ships with thousands
of astronauts aboard each. These ships would carry tens of thousands
of massive bombs, and hundreds of thousands of smaller bombs. These
smaller bombs would move the spaceship itself. The larger ones would
move asteroids.

They would fly out into the solar system, surveying all the small
bodies throughout. They would then choose the richest of these bodies
to return to Earth orbit. Once there, they would enter a stable orbit
along with all the other returning bodies, and they would form a vast
industrial feedstock for the next step in human industrial
development.

Remotely controlled factories, manned by remotely controlled robots,
would be orbited -

http://world.honda.com/ASIMO/

and people anywhere on Earth could link in and find work via a
wireless broadband internet available from space

http://www.teledesic.com/default.htm

Things would be manufactured and delivered ballistically from orbit
anywhere they're desired on Earth.

Things would also be manufactured for use on orbit - to expand the
industrial capacity of orbiting industries.

Power satellites could be cheaply built on orbit this way - and energy
beamed to Earth.

In addition to solar pumped masers we could build solar pumped lasers
to support wide scale use of laser propulsion

http://www-phys.llnl.gov/clementine/ATP/Lsrprp1s.gif

http://science.howstuffworks.com/light-propulsion.htm

This would allow people to travel ballistically throughout the world
in personal rapid transit vehicles. People could travel anywhere in
minutes without any roads, fuel, or pollution.

Large pressure vessels could be constructed in space to grow food and
forests - delivering food and fiber to people in unprecedented
quantities.

All industries on Earth could transfer their operations to space since
this would give them access to the largest markets possible at the
lowest possible price.

Ultimately, personal spaceships would make it possible for people to
visit orbit - and take up residences in their own personal space homes

http://www.permanent.com/s-index.htm

But in this visionary view of things, they wouldn't be packed in like
sardines with 100,000 others, here, they'd own stations individually -
with billions of stations on orbit simultaneously.

http://members.aol.com/oscarcombs/settle.htm

They would be organized the same way the particles of Saturn's rings
would be organized - using gravitational interaction to maintain a
safe and permanent separation among billions of interacting worlds

http://ringmaster.arc.nasa.gov/satur...r/plate_05.gif

The stable existence of Saturn's rings - consisting of trillions of
individual moonlets - is an existence proof that billions of
individual space stations each the size of a city - could all share
the same skies above Earth.


Improvements in laser propulsion - and expansion of the infrastructure
that delivers laser energy to include solar orbiting stations - will
give wing to the orbiting stations - making of them moveable homes
that can roam the solar system, and even to other solar systems, using
laser light sails.

http://library.thinkquest.org/C00376...dlife/sail.jpg
http://www.angelfire.com/space/cruis...Light_Sail.jpg

At this point humanity will begin its history as a space faring
species.

William Mook