2 Aspects of Meteoroids Flight in the Earth’s Atmosphere
15
where ρ 0 is the atmospheric density with z = 0, h is the characteristic scale of altitude.
In the Earth’s atmosphere, for heights z < 120 km, the average value of h = 7 km. To
solve the system of Eqs. 2.1–2.4, initial conditions are set for t = 0 : V = V e , θ =
θ e , L = 0, z e = 100 km.
We transform Eqs. 2.1–2.2 as follows:
dV
dt
= g sin θ −
ρV
2
2λ
,
(2.5)
dθ
dt
=
g
V
−
V
R E + z
cos θ − K
ρV
2λ
.
(2.6)
Equations 2.5–2.6 contain two aerodynamic coefficients: the ballistic coefficient
λ = M/C D S and the aerodynamic quality K = C N /C D . Moreover, the coefficient
K for meteor bodies cannot exactly be equal to zero due to the imperfection of their
shape, and its value for bodies of irregular geometric shape at hypersonic speeds can
be more than 0.1 [5]. When estimating the ballistic coefficient λ for large meteor
bodies with a mass of about 10
6 t at a density of 3 g/cm
3 , it was found that it can reach
λ = 10
5 kg/m
2 . As a result, at high altitudes, the terms in Eqs. 2.5–2.6 representing
the aerodynamic forces will be small, meaning that the atmosphere in this case has
little effect on the movement of the body. This is the peculiarity of the movement of
large meteor bodies in the atmosphere: the ability to penetrate the atmosphere.
2.3 Results of the Calculations
Using the system of Eqs. 2.1–2.4, calculations were made for a stony meteor body
with a density ρ b = 3 g/sm
3 and mass M = 1×10
6 t, entering the Earth’s atmosphere
at a speed of 30 km/s (these parameters presumably correspond to the Tunguska
meteoroid) at different initial angles of entry of the body into the atmosphere. It is
assumed that the coefficient of drag is equal to C D = 1, and the value of the ballistic
coefficient is λ ≈ 1.7 × 10
5 kg/m
2 . Figure 2.1 shows the changes in the angle
of inclination of the trajectory θ depending on the flight time for different initial
angles of entry of the body into the atmosphere θ e without taking into account the
fragmentation of the considered body and assuming zero lift (K = 0).
It can be seen that the initial angle of entry into the atmosphere has a strong
influence on the trajectory and flight time of the body. When θ e ≤ 9
◦ the angle of
the trajectory changes sign over time, and the trajectory becomes ascending. For the
angle θ e = 9
◦ “ascent” begins on the 40th s of the flight, θ e = 7
◦ and θ e = 5
◦ the
trajectory becomes ascending on the 30th s and the 20th s, respectively.
Data in Fig. 2.2 show how the height of the asteroid’s flight changes depending on
the flight time for different angles of its entry into the atmosphere. From the results
shown, it can be seen that when θ e > 9
◦ the meteorite will fall to the Earth, and
when θ e ≤ 9
◦ , starting from a certain height, its trajectory becomes ascending.
15
where ρ 0 is the atmospheric density with z = 0, h is the characteristic scale of altitude.
In the Earth’s atmosphere, for heights z < 120 km, the average value of h = 7 km. To
solve the system of Eqs. 2.1–2.4, initial conditions are set for t = 0 : V = V e , θ =
θ e , L = 0, z e = 100 km.
We transform Eqs. 2.1–2.2 as follows:
dV
dt
= g sin θ −
ρV
2
2λ
,
(2.5)
dθ
dt
=
g
V
−
V
R E + z
cos θ − K
ρV
2λ
.
(2.6)
Equations 2.5–2.6 contain two aerodynamic coefficients: the ballistic coefficient
λ = M/C D S and the aerodynamic quality K = C N /C D . Moreover, the coefficient
K for meteor bodies cannot exactly be equal to zero due to the imperfection of their
shape, and its value for bodies of irregular geometric shape at hypersonic speeds can
be more than 0.1 [5]. When estimating the ballistic coefficient λ for large meteor
bodies with a mass of about 10
6 t at a density of 3 g/cm
3 , it was found that it can reach
λ = 10
5 kg/m
2 . As a result, at high altitudes, the terms in Eqs. 2.5–2.6 representing
the aerodynamic forces will be small, meaning that the atmosphere in this case has
little effect on the movement of the body. This is the peculiarity of the movement of
large meteor bodies in the atmosphere: the ability to penetrate the atmosphere.
2.3 Results of the Calculations
Using the system of Eqs. 2.1–2.4, calculations were made for a stony meteor body
with a density ρ b = 3 g/sm
3 and mass M = 1×10
6 t, entering the Earth’s atmosphere
at a speed of 30 km/s (these parameters presumably correspond to the Tunguska
meteoroid) at different initial angles of entry of the body into the atmosphere. It is
assumed that the coefficient of drag is equal to C D = 1, and the value of the ballistic
coefficient is λ ≈ 1.7 × 10
5 kg/m
2 . Figure 2.1 shows the changes in the angle
of inclination of the trajectory θ depending on the flight time for different initial
angles of entry of the body into the atmosphere θ e without taking into account the
fragmentation of the considered body and assuming zero lift (K = 0).
It can be seen that the initial angle of entry into the atmosphere has a strong
influence on the trajectory and flight time of the body. When θ e ≤ 9
◦ the angle of
the trajectory changes sign over time, and the trajectory becomes ascending. For the
angle θ e = 9
◦ “ascent” begins on the 40th s of the flight, θ e = 7
◦ and θ e = 5
◦ the
trajectory becomes ascending on the 30th s and the 20th s, respectively.
Data in Fig. 2.2 show how the height of the asteroid’s flight changes depending on
the flight time for different angles of its entry into the atmosphere. From the results
shown, it can be seen that when θ e > 9
◦ the meteorite will fall to the Earth, and
when θ e ≤ 9
◦ , starting from a certain height, its trajectory becomes ascending.