When a meteoroid enters a planetary atmosphere there are three basic outcomes depending on the meteoroid mass and the extent of the atmosphere:
THE CONSERVATION OF MOMENTUM
The principle of momentum conservation states that the total momentum of a system remains the same in the absence of external forces. This holds in the collision between two bodies, and it holds even in the case of ineleastic collisions, even when mechanical energy is not conserved. (Of course, total energy is conserved, but some of the kinetic energy is converted into heat energy).
This principle can be illustrated by an apparatus known irreverently as Newton's balls:
These devices can often be purchased from popular science shops or museums. They normally consist of four to six metal balls, of identical masses, suspended from a frame by nylon or other inelastic line.
For the purposes of this demonstration we only need to consider two of the balls. If we raise the left hand ball a small distance (keeping the strings taught) and then release it, the gravitational potential energy we gave it, by virtue of increased height, will be converted into kinetic energy, and it will collide with the right hand ball with a speed v metres per second. The left hand ball comes to an immediate stop while its partner moves off to the right with the same speed v.
We may summarise this experiment simply by saying that a body will be brought to rest through collision with a body of the same mass.
Mathematically, we can write:
if and only if m_{1} = m_{2}
APPLICATION TO ATMOSPHERIC ENTRY OF A METEOROID
The application of momentum conservation to a meteoroid is that a meteoroid will be brought to rest when it has encountered its own mass of atmosphere.
A planetary atmosphere can be approximated by the negative exponential formula:
We now make a bigger assumption by replacing the real atmosphere with a scale atmosphere. This scale atmosphere has a uniform density with height from the surface up until a height h=H. We find that such a scale atmosphere has an identical total mass as does the real atmosphere. The mathematical description of the scale atmosphere is then:
VERTICAL INCIDENCE
We now use the scale atmosphere to consider the entry of a meteoroid into the atmosphere at vertical incidence (ie coming straight down, perpendicular to the planetary surface).
Consider a spherical meteroid of diameter d and mass m_{m} travelling at vertical incidence from the top of the scale atmosphere down toward the surface. It will travel through a cylindrical column of atmosphere. It will be brought to rest when (or if) the mass of the atmosphere encountered in the column equals its own mass. That is: |
where V_{a} & V_{m} are the volumes of the atmosphere encountered and of the meteoroid respectively.
The maximum size meteoroid that will be brought to rest in travelling through the atmospheric column of length H can be found by substituting the values for the atmospheric column and meteoroid volumes:
GRAZING INCIDENCE
The most extreme case for a meteoroid atmospheric entry is a grazing entry. Essentially this gives the largest amount of atmosphere available for braking. In the diagram C is the centre of the planet, R is its radius, H the height of the scale atmosphere and x is the length of the atmospheric column that the meteoroid may traverse. |
Applying the pythagorean theorem to the right angled triangle we have:
In practice we find this is an overestimate, and the maximum column of atmosphere traversed in the most extreme grazing entry is closer to 10 times (rather than 36 times) the height of the scale atmosphere.
This means that in such an entry the atmosphere can brake a meteoroid ten times the diameter of one at vertical incidence and 1000 times the mass (mass scales as distance cubed).
SPECIFICS
We can apply the above results to find maximum diameters and masses of fully braked meteoroids for a particular planet and meteoroid type.
For the Earth, where the maximum atmospheric density ios about 1 kg/m^{3} and the scale height near the surface is around 10 km. Two types of meteoroids are commonly incident to the Earth. Stony bodies with a density of 3500 kg/m^{3} and iron bodies with a density of about 8000 kg/m^{3}. The table below shows the maximum braked diameter (metres) and mass (tons) for both vertical and grazing incidence.
Incidence | STONE | IRON |
Vertical | 4m / 100t | 2m / 30t |
Grazing | 40m / 100,000t | 20m / 30,000t |
Meteoroids much larger than the sizes specified in the table will hit the Earth with substantial velocities forming a crater and consuming most of the meteoroid in the process.
Meteoroids with much smaller sizes than those indicated will generally be consumed in the ablation process as they exchange mass for kinetic energy reduction.
Meteoroids a little smaller than the maximum sizes indicated will not be completely consumed (ablated) by atmospheric entry and will drop a meteorite, which falls to the Earth at a terminal velocity of around 100 metres per second. This is much less than the space velocity in excess to 10km/s that the meteoroid has when it first encounters the Earth's atmosphere. Normally the initial mass of a meteoroid has to exceed somewhere between about 100 to 1000 kg to leave a meteorite on the ground.
Australian Space Academy