The calculations, as promissed:
Ok. I'll assume we're taking the approach of breaking the asteroid into lots of small pieces. The reasons for this are the following:
1. It would take much more drilling work than is required to alter the asteroid's orbit (since more matter needs to be chipped away). Hence this will give a good upper limit to the time required for the operation.
2. The required systems can be much simpler, since no matter needs to be accellerated for thrust.
3. It is simpler to analize, since there are no orbital considerations.
I will also assume nanotechnology is used, since I am more familiar with the area than with MEMS. However, the two are quite similar, and the calculation below should be easy to adapt for the MEMS case. Further, I am assumming an asteroid and not a comet, since comets are generally weaker and AFAIK also smaller than (the largest) asteroids, so they should be easier to deal with.
First, some raw data and facts:
- The typical operating frequency of a (high performance) nano-device, is on the order of 1 GHz or more [1,2].
- The typical diameter of a complex nano-robotic device is (expected to be) around 100-1000 nm.
- The expected structural materials for nano-robotic devices are carbon or silicon based, most likely diamondoid.
- The density of diamond is 3.51 gm/cm^3 and that of graphite is 2.26 gm/cm^3 [3]. These correspond to 3510 and 2260 kg/m^3, respectively.
- The density of silicon is 2330 kg/m^3 [4].
- The largest known asteroid (or maybe 2nd largest [6]) is Ceres, and has a mass of about 4.35e-10 solar masses [5], which is about 8.66e20 kg (solar mass value taken direcly from google as 1.99e30 kg).
- Ceres has a diameter of about 622 miles [7], or around 1000 km (conversion done with
http://www.onlineconversion.com).
- The largest known near-earth asteroid is 1036 Ganymed, about 41 km in diameter [8].
- The mass of the NEAR Shoemaker probe was 818 kg [9].
- The cell parameter of an iron crystal is 286.65 pm [10], which is about 2.87e-10 m.
- The cell parameter of a nickel crystal is 352.4 pm [11], which is about 3.52e-10 m.
Given the above data, I will assume the following values for the calculation:
- Nanite operating frequency:
Fn = 5e7 Hz = 50 MHz (a fairly modest value).
- Nanite radius:
Rn = 2.5e-7 m = 250 nm (500 nm diameter).
- Nanite density:
Dn = 3000 kg/m^3.
- Asteroid radius:
Ra = 1e5 m (approximately worst-case scenario, 200 km diameter).
- Asteroid average cell parameter, assumming M-type (pure nickel/iron), which is more dangerous:
La = 3e-10 m.
- Average functioning nanite mass (allowing for medium failure rates):
Mn = 500 kg.
- Desired particle radius:
Rp = 0.5e-3 m = 1 mm (small enough to be harmless, except perhaps for satellies).
For simplicity, I'll assume the nanites, asteroid and so on are all perfect spheres. The equtions are then:
Volume of a sphere:
(1)
V = 4/3 * pi * R^3
Surface area of a sphere:
(2)
S = 4 * pi * R^2
Area of a circle:
(3)
A = pi * R^2
Number of particles that need to be produced (a - asteroid; p - particle):
Np = Va / Vp = (4/3 * pi * Ra^3) / (4/3 * pi * Rp^3)
(4)
Np = (Ra/Rp)^3
Volume of matter that needs to be grinded away to break up the asteroid into the desired particles - number of particles times particle surface area times nanite radius:
Vm = Np * Sp * Rn = (Ra/Rp)^3 * 4 * pi * Rp^2 * Rn
(5)
Vm = 4 * pi * Rn * Ra^3 / Rp
Average number of active nanites - total mass over nanite mass:
Nn = Mn / (Vn * Dn) = Mn / (4/3 * pi * Rn^3 * Dn)
(6)
Nn = 3 * Mn / (4 * pi * Dn * Rn^3)
Assuming each nanite removes one atomic layer per cycle, over an circular area with radius Rn, the volume-per-second of matter removed by all the nanites is:
VPS = An * La * Fn * Nn = pi * Rn^2 * La * Fn * 3 * Mn / (4 * pi * Dn * Rn^3)
(7)
VPS = 3 * La * Fn * Mn / (4 * Dn * Rn)
And finally, the time it would take to finish the job:
t = Vm / VPS = {4 * pi * Rn * Ra^3 / Rp} / {3 * La * Fn * Mn / (4 * Dn * Rn)}
t = 4 * pi * Rn * Ra^3 * 4 * Dn * Rn / (3 * La * Fn * Mn * Rp)
(8)
t = 16 * pi * Dn * Rn^2 * Ra^3 / (3 * La * Fn * Mn * Rp)
Using (8) with the above values, we get:
t = 8.38e8 s = ~9700 days = ~26.57 years
Although this sound like much, this is actually not that bad considering the size of the asteroid and the comparitively small total mass of nanites. At this rate, it would take only about 80 days to decompose 1036 Ganymed if it was of similar composition (and roughly spherical, or of equivalent volume - what exactly does an asteroid "diameter" signify, anyway?). Changing its orbit would take less time, since only a fraction of the mass would need to be chipped away.
Btw, considering the final form of the equation, it is easy to calculate the result for MEMS, we just need to know the MEMS' radius, density, and operational frequency... and also how thick a layer of matter they can remove per cycle (i.e. the equivalent value for La). Anyone has some sample values?
References:
[1]
http://www.nanomedicine.com/NMI/2.3.2.htm
[2]
http://www.nanomedicine.com/NMI/2.4.1.htm
[3]
http://hyperphysics.phy-astr.gsu.edu/hbase/minerals/diamond.html
[4]
http://www.webelements.com/webelements/elements/text/Si/phys.html
[5] http://aa.usno.navy.mil/ephemerides/asteroid/astr_alm/asteroid_ephemerides.html
[6] http://www.planetary.org/html/news/articlearchive/headlines/2001/2001kx76.html
[7] http://www.seasky.org/solarsystem/sky3k.html
[8]
http://www.nasm.si.edu/research/ceps/etp/asteroids/AST_near.html
[9]
http://space.skyrocket.de/index_frame.htm?http://space.skyrocket.de/doc_sdat/near.htm
[10]
http://www.webelements.com/webelements/elements/text/Fe/xtal.html
[11]
http://www.webelements.com/webelements/elements/text/Ni/xtal.html