Razi Abid said:
I have seen many movies relating to the destruction of the Earth from space rocks and asteroids. Sometimes I think our planet might be attacked by such things. Even more I heard that scientist have observed an asteroid heading to Earth and might crash after about 800 years. Can this ever happen?
800 years!? I'd like to know how anybody can accurately predict the cumulative perturbations of an Earth-crossing asteroid across that much time...accurately enough to predict a collision or a miss with Earth, that is.
I remember that JPL had to do a probability distribution for a 2001 XU encounter, and that wasn't anywhere near eight centuries. (2001 XU is an asteroid in an 8:33 tidal resonance with Earth, with a closest possible approach - for now - of 19,000 kilometers.)
An asteroid with an absolute magnitude (solar system scale) of 22.0 and a (typical) surface albedo of 0.13 will be about 150 meters in diameter. A difference of one magnitude corresponds to an increase or decrease in reflected power by the fifth root of 100, or 2.511886431.
For example, 2001-XU has an absolute magnitude of 19.01, and therefore...
Reflected power = 100^{(22-19.01)/5} = 15.7
...as compared to the reflected power from a 150-meter asteroid.
Reflected power varies according to surface area, which depends on the square of the average radius. So 2001-XU is about 3.96 times as wide as a 150-meter asteroid is. Its average radius is about 297 meters.
If we pretend that the asteroid is spherical, then the volume is 110 million cubic meters.
If we assume that the average density is 3000 kilograms per cubic meter (typical for rock), the mass is about 330 billion kilograms. (No attempt at precision is being made here; this is just a ballpark calculation.)
Here's an equation that I put together to let you reach the impact energy directly.
K = 1.26 megatons TNT { 100^[0.3 (22-MAG)] } V^2
Where K is kinetic energy, MAG is the asteroid's absolute magnitude, and V is the asteroid's speed relative to Earth at impact, in kilometers per second.
The equation implicitly assumes that the asteroid reflects 13% of incident sunlight. You would discover the relative speed by reducing the orbital elements for Earth and for the asteroid to obtain the velocity vectors of each at the moment of impact, viz...
*** top of velocity calculation section ***
a : semimajor axis of the orbit
e : eccentricity of the orbit
1 astronomical unit = 1.49597870691E+11 meters
GMsun = 1.32712440018E+20 m^3 sec^-2
The canonical velocity in a hyperbolic orbit, in general, is found from
Vx’’’ = -(a/r) { GMsun / a }^0.5 sinh u
Vy’’’ = +(a/r) { GMsun / a }^0.5 (e^2 - 1)^0.5 cosh u
Vz’’’ = 0
Where u is the eccentric anomaly and r is the current distance from the sun.
The canonical velocity in an elliptical orbit, in general, is found from
Vx’’’ = -sin Q { GMsun / [ a (1-e^2) ] }^0.5
Vy’’’ = (e + cos Q) { GMsun / [ a (1-e^2) ] }^0.5
Vz’’’ = 0
Where Q is the true anomaly.
The triple-primed vectors would be rotated (negatively) by the angular elements of the orbit (w,i,L) to heliocentric ecliptic coordinates.
Rotation by the argument of the perihelion, w.
Vx'' = Vx''' cos w - Vy''' sin w
Vy'' = Vx''' sin w + Vy''' cos w
Vz'' = Vz''' = 0
Rotation by the inclination, i.
Vx' = Vx''
Vy' = Vy'' cos i
Vz' = Vy'' sin i
Rotation by the longitude of ascending node, L.
Vx = Vx' cos L - Vy' sin L
Vy = Vx' sin L + Vy' cos L
Vz = Vz'
The unprimed vector [Vx, Vy, Vz] is the velocity in the orbit, referred to heliocentric ecliptic coordinates.
*** end of velocity calculation section ***
The impact speed will be the magnitude of the vector difference between the HEC velocity of the asteroid and the HEC velocity of the Earth.
If 2001-XU ever hits Earth, it will be moving at 28.9 km/sec, which corresponds to an impact energy of 2.76+20 Joules, which is equivalent to 65,600 megatons TNT (or 65.6 gigatons). This kind of thing might, possibly, you know, destroy the world.
Modeling asteroid impact effects.
http://www.lpl.arizona.edu/impacteffects/
Asteroid orbital elements.
http://ssd.jpl.nasa.gov/sb_elem.html
http://arnold.usno.navy.mil/murison/Asteroids/OrbitalElements.html
Jerry Abbott
