The Atmosphere as a Meteoroid Brake

CONSERVATION OF MOMENTUM

A moving body will be brought to rest if it collides with another body of equal mass. The second body will then move off with the same velocity as the first body. This is demonstrated by the classic experiment with two suspended equal size balls. The law involved here in the conservation of momentum.

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

If the two masses are equal ( m₁ = m₂ ) and mass 2 is initially at rest ( u₂ = 0 ) then after the collision mass 1 will be at rest and mass 2 will move off with the velocity mass 1 had before impact (ie v₂ = u₁ ).

This law also applies when a meteoroid enters the Earth's atmosphere. It will be brought to rest after it encounters an equal mass of air. This can be used to determine the maximum size of a meteorite. Any larger meteoroids will still have some of their hypervelocity and will create a crater and destroy themselves on impact.

Thus the maximum size body will just be brought to rest when:

m_atmos = m_body or

ρ_a V_atmos = ρ_m V_m

Now:

V_atmos> = π ( d / 2 )² H

V_m = ( 4 / 3 ) π ( d / 2 )³

Equating the two masses and solving for diameter gives:

Diameter dmax = (3/2) H ρa / ρm . . . Vertical Incidence

The mass can be solved from the relation m_max = ρ_m V_m, and this gives:

Mass mmax = (9/16) π H3 ρa3 / ρm2 . . . Vertical Incidence

ATMOSPHERIC SCALE HEIGHT

A rough approximation for the Earth's atmospheric density at a height h is given by the formula:

ρ(h) = ρ_o e^-h/H

where ρ_o is the density at ground level and H is called the scale height. It is called this because a column of air of height H and uniform density ρ_o has the same mass of air as an infinite column of air with exponentially decreasing density as given by the above equation. This can be summed up by saying:

Real atmosphere: ρ(h) = ρ_o e^-h/H

Scale atmosphere: ρ(h) = ρ_o ( for h < H ) .and. ρ(h) = 0 ( for h > H )

The average scale height for the Earth's atmosphere below 100 km altitude is around 7000 m or 7 km.

GRAZING INCIDENCE

For grazing incidence we have:

G² + R² = ( H + R )²

X² = ( H + R )² - R²

X² ≈ 2 H R since H² << R

This gives:

X = √ ( 2 H R )

X = 300
= 43 H

The equivalent airmass however is only about 12 H (this can be shown by integration of the exponential atmosphere.

NUMBERS

The Earth's atmosphere has a surface density of ρ ≈ 1 kg/m³ and a scale height of around 7 km.

A stony meteoroid has a density ρ_m of ~ 3500 kg/m³ and an iron meteoroid a density ρ_m ~ 8000 kg/m³.

Substituting these numbers in the previous equations we have:

STONE METEORITE

Maximum Diameter d ~ 3 m

Maximum Mass m ~ 50 ton

IRON METEORITE

Maximum Diameter d ~ 2.3 m

Maximum Mass m ~ 49 ton

For a low angle entry, these figures need to be multiplied by ~ 10 for the diameter and ~ 1000 for the mass. No meteorite anywhere approaching these dimensions has ever been found. Fragmentation is one possible explanation.

In the real world we find that the largest intact meteorite ever found had a 'size' of maybe one metre and an estimated mass of over 60 tons. It was found in Namibia in Africa and was named the Hoba meteorite after the station where it was found. It is an iron meteorite.

Australian Space Academy