A&A 365, 612-621 (2001)
DOI: 10.1051/0004-6361:20000154
L. Foschini
Send offprint request: L. Foschini,
Institute TeSRE - CNR, Via Gobetti 101, 40129 Bologna, Italy
Received 22 August 2000 / Accepted 18 October 2000
Abstract
It is known, from observational data recorded from airbursts, that
small asteroids breakup at dynamical pressures lower than their
mechanical strength. This means that actual theoretical models are
inconsistent with observations. In this paper, we present a detailed
discussion about data recorded from airbursts and about several
theoretical models. We extend and improve a theory previously
outlined for the fragmentation of small asteroids in the Earth
atmosphere. The new condition for fragmentation is given by the
shock wave-turbulence interaction, which results in sudden outburst
of the dynamical pressure.
Key words: minor planets, asteroids - plasmas - shock waves
Author for correspondance: foschini@tesre.bo.cnr.it
Studies on the fragmentation of small asteroids have also a great importance in impact hazard. Although the damage caused by Tunguska-like objects can be defined as "local'', it is not negligible. The Tunguska event of 30 June 1908 resulted in the devastation of an area of km^{2} and the destruction of more than 80 million trees (for a review, see Trayner 1997; Vasilyev 1998). Still today there is a wide debate all over the world about the nature of the cosmic body which caused that disaster. Just on July 1999 an Italian scientific expedition, Tunguska99, went to Siberia to collect data and samples (Longo et al. 1999).
In this article, we discuss firstly the current models on atmospheric fragmentation of small asteroids and inconsistencies with observations (Sect. 2). In Sect. 3, we deal with the question of strength and aerodynamic load. In Sect. 4, we extend the approach previously outlined in Foschini (1999b), thereafter called Paper I. We study the hypersonic flow around a small asteroid, with particular reference to the definition of the pressure crushing on a cosmic body (Sect. 5). Therefore (Sect. 6), it is possible to obtain the condition for fragmentation under steady state conditions, that was already outlined in the Paper I. The Sect. 7 deals with the turbulence and massive ablation, putting the basis for the condition of fragmentation under unsteady conditions (Sect. 8). Some numerical examples conclude the paper (Sect. 9).
However, observations of very bright bolides prove that large meteoroids or small asteroids breakup at dynamical pressures lower than their mechanical strength. Today there is still no explanation for this conundrum. This is a scientific problem of great interest, but it is also of paramount importance, because it allows us to know whether or not an asteroid might reach the Earth surface. In addition to this, the atmospheric breakup has also effect on the crater field formation (Passey & Melosh 1980) or on the area devastated by the airblast.
The interaction of a cosmic body in the Earth atmosphere in the strength-dominated regime can be divided into two parts, according to the body dimensions. For millimetre- to metre-sized bodies (meteoroids), the most useful theoretical model is the gross-fragmentation model developed by Ceplecha et al. (1993) and Ceplecha (1999). In this model, there are two basic fragmentation phenomena: continuous fragmentation, which is the main process of the meteoroid ablation, and sudden fragmentation or the discrete fragmentation at a certain point.
For small asteroids another model is used, where the ablation is contained in the form of explosive fragmentation, while at high atmospheric heights it is considered negligible. Several models have been developed: Baldwin & Shaeffer (1971), Grigoryan (1979), Chyba et al. (1993), Hills & Goda (1993), Lyne et al. (1996). A comparative study on models by Grigoryan, Hills & Goda, and Chyba-Thomas-Zahnle was carried out by Bronshten (1995). He notes that the model proposed by Chyba et al. does not take into account fragmentation: therefore, the destruction heights are overestimated (about 10-12 km). Bronshten also concludes that Grigoryan and Hills-Goda's models are equivalent.
Despite the particular features of each model, fragmentation is always considered to start when Eq. (1) is satisfied.
Although direct observations of asteroid impact are not available, it
is possible to compare these models with observations of bodies with
dimensions of several metres or tens of metres. Indeed, in this
range, the gross-fragmentation model overlaps the explosive
fragmentation models. As underlined several times by Ceplecha (1994,
1995, 1996b), observations clearly show that meteoroids breakup at
dynamical pressures lower (10 times and more) than their mechanical
strength. These data are obtained from photographic observation of
bright bolides and the application of the gross-fragmentation model,
that can be very precise.
Category | Range of [MPa] | Mean [MPa] |
a | 0.08 | |
b | 0.25 | |
c | 0.53 | |
d | 0.80 | |
e | 1.10 |
According to Ceplecha et al. (1993) it is possible to distinguish five strength categories with an average dynamical pressure of fragmentation (Table 1). For continuous fragmentation the results obtained also indicate that the maximum dynamical pressure is below 1.2 MPa. Five exceptions were found: 4 bolides reached 1.5 MPa and one survived up to 5 MPa (Ceplecha et al. 1993).
It would be also very important to relate the ablation coefficient with the dynamical pressure at the fragmentation point, in order to find a relationship between the meteoroid composition and its resistance to the air flow. To our knowledge, a detailed statistical analysis on this subject does not exist, but in the paper by Ceplecha et al. (1993) we can find a plot made by considering data on 30 bolides (we refer to Fig. 12 in that paper). We note that stony bodies (type I) have a wide range of values at the fragmentation point. In the case of weak bodies, we can see that there is only one cometary bolide (type IIIA), but this is due to two factors: firstly, cometary bodies undergo continuous fragmentation, rather than a discrete breakup at certain points. Therefore, it is incorrect to speak about fragmentation pressure; we should use the maximum tolerable pressure. The second reason is that there is a selection effect. Indeed, from statistical studies, Ceplecha et al. (1997) found that a large part of bodies in the size range from 2 to 15 m are weak cometary bodies. However, a recent paper has shown that statistics from physical properties can lead to different results when compared with statistics from orbital evolution (Foschini et al. 2000). To be more precise, physical parameters prove that, as indicated above, a large part of small near Earth objects are weak cometary bodies, whilst the analysis of orbital evolution proves a strong asteroidal component.
In addition to data published in the paper by Ceplecha et al. (1993) and Ceplecha (1994) we can see in Table 2 some specific cases of bright bolides; for details, see the papers quoted or to Foschini (2000).
Name | Date | S | Source | |
Prìbram | Apr. 7, 1959 | 9.2 | 50 | ReVelle (1979) |
Lost City | Jan. 3, 1970 | 1.5 | 50 | Ceplecha (1996a) |
Sumava | Dec. 4, 1974 | 0.14 | 1 | Borovicka & Spurný (1996) |
Innisfree | Feb. 6, 1977 | 1.8 | 10 | ReVelle (1979) |
Space based obs. | Apr. 15, 1988 | 2.0 | 50 | Nemtchinov et al. (1997) |
Space based obs. | Oct. 1, 1990 | 1.5 | 50 | Nemtchinov et al. (1997) |
Benesov | May. 7, 1991 | 0.5 | 10 | Borovicka et al. (1998a,b) |
Peekskill | Oct. 9, 1992 | 1.0 | 30 | Ceplecha et al. (1996) |
Marshall Isl. | Feb. 1, 1994 | 15 | 200 | Nemtchinov et al. (1997) |
The scaling law for mechanical strength derives from the assumption that the fragmentation was a process of consecutive elimination of defects under increasing load. Baldwin & Sheaffer (1971) consider that the reason for the presence of cosmic bodies with very low fragmentation pressure can be explained by the assumption that additional flaws and cracks may be created by collisions in space, even though they do not completely destroy the cosmic body.
Tsvetkov & Skripnik (1991) made an interesting study on the fragmentation according to strength theory and scaling laws, but this kind of study is useful only for meteorites and cannot explain airbursts. They also show that the condition of fragmentation is not valid even under the assumptions of scaling laws. Indeed, they showed that the aerodynamic loading never reach the ultimate strength. For this reason, they searched the cause of fragmentation in the particular structure and extreme inhomogeneity of cosmic bodies. With an appropriate selection of the scale factor, Tsvetkov & Skripnik obtained a good fit.
However, the shock compression and heating during the atmospheric entry will result in an elimination of internal cracks, making the body more compact (see, for example, Davison & Graham 1979 or Zel'dovich & Raizer 1966). If internal voids are so large to survive to shock compression, they could give an explanation for some episodes, but not a general theory (Foschini 1998).
If we turn our attention to the hypersonic flow around the body, we
could have data from direct observations. Among models discussed
above, only Nemtchinov et al. (1997) tried to investigate the
hypersonic flow around the asteroid with a numerical model. Foschini (1999b) investigated the analytic approach: indeed, although the
details of hypersonic flow are very difficult to calculate and
there is need of numerical models, the pressure distribution can be
evaluated with reasonable precision by means of approximate methods.
In the limit of a strong shock ()
several equations tend to
asymptotic values and calculations become easier. For example, the
ratio of
densities across the shock is:
where is the density in the stagnation point and is the specific heats ratio.
In the Paper I, it was showed the crucial role of the temperature, instead of the pressure, in the stagnation point. The gas is in local thermodynamic equilibrium (LTE), that is, matter is in equilibrium with itself, but not with radiation, which can escape. Particle densities depend on the temperature only and it is possible to use Boltzmann or Saha equations. The temperature is calculated directly from observations of spectra (e.g. Borovicka & Spurný 1996; Borovicka et al. 1998b) and, thus, it is a reliable starting point.
Particularly, as stated in the Paper I, the temperature at the stagnation
point is very important. Changes in the stream properties are mainly due
to changes in the stagnation temperature, which is a direct measure of
the amount of the heat transfer. The enthalpy change
is:
Under certain approximations, the flow can be considered adiabatic: heat transfer in hypersonic motion is strongly reduced. The boundary layer, where convective heat transfer takes place, is very thin and can be considered negligible.
On the other hand, if we consider the Eq. (5), we see that it is satisfied for unrealistic high temperatures. For example, for a speed of V=12 kms^{-1} in air we obtain K. The introduction of ablated matter make higher and R lower, resulting in a further increase of T_{0}.
However, at high temperatures some effects of the gas become important: the main effects are the dissociation and ionization of air molecules. For 2000<T<4000 K the oxygen molecules break down to single atoms; for 4000<T<9000 K there is the dissociation of N_{2}; for T>9000 K ionization occurs (see, for example, Oosthuizen & Carscallen 1997). In the term of the Eq. (4), we should take into account also the heat absorbed by these processes. As a result, T_{0} drops to more reasonable 10000-15000 K or so.
The treatment of real gas properties is probably one of the most difficult problem in aerothermodynamics and involves complex numerical modelling (for a review, see Tirsky 1993; Gnoffo 1999).
From the point of view of pressure, we can see that the gas around an asteroid during the atmospheric entry, is still an ideal gas. Indeed, the limit of pressure to become a degenerate gas is around 10^{14} Pa (Eliezer 1991). A simple calculation shows that the asteroid reaches the maximum dynamical pressure at the sea level ( kg/m^{3}) and if it has the maximum geocentric speed (V=72 kms^{-1}, although it is quite impossible to have an asteroid with such a speed). Therefore, the maximum Pa, about four orders of magnitude below the limit of degeneracy.
From the point of view of the temperature, the question is a bit difficult to handle. From spectroscopic observations of bright bolides (Borovicka & Spurný 1996; Borovicka et al. 1998b), we can see that the spectrum is composed by two temperatures emission: the first value lies in the range between 4000-6000 K and the second value is about 10000 K. The temperature of 10000 K is about the limit between normal and partially ionized gas. Therefore, we shall adopt a more general point of view, based on microscopic physics.
From the kinetic theory, we know that the pressure is additive, therefore
we can separate it in contributions from different species composing the gas:
electrons, photons, ions, and atoms. We can write:
where k is the Boltzmann constant and a is the radiation constant, related to the Stefan-Boltzmann constant ( ); N is the volume density of species (the index e is for electrons); the sum over the index j concerns all heavy particles (ions and neutral atoms).
Equation (6) has been obtained by means of quite general assumptions, without any particular restrictions (more details can be found in many places; for example, in Lang 1999). We can apply Eq. (6) to hypersonic flow around an asteroid in the Earth atmosphere. The only restrictions are determined by the momentum distributions of particles: ions, atoms, and electrons obey generally to a Maxwellian distribution, photons obey to the Bose-Einstein statistics. These conditions are satisfied in processes occurring during the atmospheric entry of a cosmic body.
where
N_{a} is the density of neutral atoms,
E_{i} is
the ionization potential,
g_{a} and
g_{i} are respectively the
statistical weight of the ground state of the neutral atom and of the
ion. Some of these values are listed in Table 3.
Species | E_{i} [eV] | g_{a} | g_{i} |
Na | 5.14 | 2 | 1 |
K | 4.34 | 2 | 1 |
O_{2} | 12.05 | 3 | 4 |
N_{2} | 15.6 | 1 | 2 |
For each airburst, with a given composition, we can calculate densities of all species and obtain the effective pressure. But this is not the purpose of this work. We will introduce some hypoteses for the sake of the simplicity, in order to do some calculations of order of magnitude.
Figure 1: Temperature as a function of the height, for different latitudes and seasons. Data from US Standard Atmosphere 1976 | |
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The Saha Eq. (8) can be rewritten in terms of degree
of ionization :
where
N_{tot} is the total number density of particles;
therefore,
and so on. The
Eq. (7) can be rewritten:
that is the well known equation of state for a gas with a degree of ionization .
Foschini (1999a) underlined the importance of alkaline metals in the production of electrons in meteors. In addition, von Zahn et al. (1999) recently showed the particular role of the potassium in Leonid meteors. Indeed, alkaline metals, owing to their low ionization potential easily ionize even at low temperatures. For example, we assume that N_{tot}=6.022 10^{23} m^{-3} (it is the Avogadro number) and we calculate the degree of ionization of a gas composed by single species listed in Table 3. For a temperature K, we have: for sodium, ; for potassium, ; for oxygen, and for nitrogen, .
It is known that, in meteors, the impact ionization from impinging molecules is very important. A nitrogen molecule with a speed of 18 kms^{-1} has an energy of about 45 eV. However, as the asteroid penetrates deeper in the atmosphere, the flow generates a shield that protects the asteroid from direct impacts (for a detailed description see Ceplecha et al. 1998). Therefore, the thermal ionization become important particularly in small asteroids/comets, when impact ionization is strongly reduced. So, alkaline metals play an important role, because they provide a huge amount of electrons.
We have also to take into account that we are dealing with particular
condition in the stagnation point, where there is the maximum
thermo-mechanical stress. In this point, we can consider the fluid
as highly ionized: the ablation of alkaline metals provide a source
of ions for the flow. In the stagnation point - not in the whole
flow - we can consider the gas with a degree of ionization
.
Therefore, by substituting Eq. (10) in
Eq. (5), we obtain the condition for fragmentation:
It is worth noting that Eq. (11) is the condition of
fragmentation, not the condition for the airburst, as deduced by
Bronshten (2000). The airburst generally occurs after a scale
height, as shown by several studies on superbolides. In the Paper I, there
was an unfortunate error in considering the explosion height, instead
of the fragmentation height (although it has negligible effect, as we
shall see in the Sect. 9).
Therefore, Eq. (11) must be revised as follows:
where is the air density at sea level, h is the airburst height, and H is the scale height.
Before to analyse the Eq. (12), it is necessary to look at the value for . In the Paper I, was used and it was taken from experimental measurement in hypervelocity impacts. Bronshten (2000) does not agree with this value and invokes argument to support a value of , that is compatible with air at high temperatures and under shock loading. Bronshten's assumption are reasonable when we have a diatomic gas at high temperature: the rotation and vibration of air molecules change the values for . But, if we have a monatomic gas or metal vapors, we have to consider a , unless the temperature is so high to change appreciably the electron energy (see Zel'dovich & Raizier 1966); however, this is not our case. This is confirmed also by meteor spectra: they show that the surrounding gas is monatomic, although molecules bands are sometime recorded (Ceplecha et al. 1998).
It is worth noting that the question about the value of is not simple. Perhaps, it will be solved when experimental data will be available.
Figure 2: Local sound speed from the reference temperature of the US Standard Atmosphere 1976 | |
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The influence of the turbulence in the large meteoroid entry was first analyzed by ReVelle (1979). In his approach based on the motion of a single body in the atmosphere, ReVelle studied how the convective heat transfer depends on turbulence. He found that the turbulent convective layer is negligible, except for speed lower than about 20 kms^{-1}.
However, the presence of massive ablation changes the flow. Gupta (1983) noted that there are few experimental data available and, most important, they are obtained with freestream Mach number in the range 3-7 and with negligible ablation. There are less data about experiments with moderate ablation, but with lower Mach number (up to 2.6). Starting from these data, models predict that convective heating is negligible with almost any turbulence model. This was the situation in 1983, but more recent reviews (Gnoffo 1999) show that there are still several things unknown. The only news is the entry of the Galileo probe in the Jovian atmosphere, which gave us useful information about hypersonic motion with massive ablation. The probe entered in the atmosphere at a relative speed of 47.5 kms^{-1} and experienced an ablation rate of 7.4 kms^{-1}, with a total mass loss of about 79 kg (Gnoffo 1999).
The probe did not suffer any fragmentation, despite of massive ablation, although it is still not clear the role of all processes during the entry. The layer between the shock and the probe is expected to be turbulent over almost the entire length of the Galileo probe, owing to the massive ablation and the large Reynolds number (Gupta 1983). The interaction of shock waves with turbulence leads to amplification of speed fluctuations and changes in length scales. It would be better to say that this occur when the flux is unsteady and the shock wave is subject to strong distortions. At high Mach number, but steady state motion, the effect of compressibility takes place and there is no amplification; we can adopt Eq. (12).
But the atmospheric motion is generally unsteady. Figure 1 shows the air temperature from tables of the US Standard Atmosphere 1976. The air temperature is instrumental to derive the local sound speed ( ), as a function of the height, for different latitude and seasons. As the temperature changes, the sound speed changes. The US Standard Atmosphere gives also a reference atmosphere - and therefore a reference temperature - from which we can calculate a reference sound speed (Fig. 2).
Figure 3: Mach number as a function of the height for different asteroid speed. M1=12 kms^{-1}; M2=40 kms^{-1}; M3=72 kms^{-1} | |
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It is known that, for a small asteroid, the deceleration due to ablation is negligible; therefore we can consider that the body speed does not change until the fragmentation begins. For a given asteroid, the Mach number depends only on the local sound speed. In Fig. 3 we can see the Mach number for different values of the speed of the cosmic body. It results that the Mach number change substantially for higher asteroid speed, while for lower speed, changes of M become smooth. We note also strong variations when the body crosses the mesopause, the stratopause, and the tropopause.
Ceplecha (1994) gave the mean values of end heights for large meteoroids (up to about 7 m): he found 32 km for type I; 43 km for type II; 57 km for type IIIA; and 69 km for type IIIB. These are end heights and if we consider that the fragmentation begins about a scale height above, we infer that the mean fragmentation height is: 39 km for type I; 50 km for type II; 64 km for type IIIA; and 77 km for type IIIB. That is, we can divide these bodies into two categories: a first category, made with type II and type IIIB, which contains bodies that breakup during the crossing of the stratopause and the mesopause, respectively. The second category, made with type I and type IIIA, which contains bodies that breakup with a certain delay after the crossing of atmospheric pauses. The delay can be explained with the different ability to ablate, and therefore to produce the turbulent boundary layer. Indeed, for bodies that breakup at the mesosphere (cometary bodies), the type IIIB has an average ablation coefficient higher than the type IIIA. For asteroidal bodies, that breakup at the stratosphere, the type II has a higher ablation coefficient than the type I. We should also take into account that the transition from laminar to turbulent flow depends on the body dimensions and speed. Large bodies at high speed reach the transition at higher heights (ReVelle 1979).
It is worth noting that there are also other factors that can change the fragmentation height, such as the rotation (Adolfsson & Gustafson 1994; Ceplecha 1996a), but they are independent from the body type and can explain only deviations from the average values.
Despite of differences, it is clear that the shock wave-turbulence interaction produce an amplification of fluctuations. The amplification depends on the shock strength, the state of the turbulence, and its level of compressibility. The most important outcomes are the amplification of velocity fluctuations and changes in the length scales (Andreopulos et al. 2000). This leads to changes in the dynamical pressure in the front of the asteroid, but also changes in pressure along the flank of the cosmic body; these changes can be further amplified by local irregularity, such as small "hills''.
However, for the sake of simplicity, we make the assumption that the
zone of greatest stress is the stagnation point. Or, it would be
better to say, that we will continue to hold this hypotesis. Being
under unsteady conditions we cannot apply isentropic relations, i.e.
Eq. (12). We start from the value of the pressure in the
shock layer derived from the Rankine-Hugoniot relations in the
hypersonic limit (
). Being the gas cap
around the asteroid a plasma, we have to consider the
Eq. (10), so we have:
In conclusion, we can assume for kinetic energy
and that
this amplification value is valid also for pressure. Therefore, under
distortion of the shock wave, the turbulence can leads to amplification
of dynamical pressure up to 12 times the nominal value for a neutral
gas (we have taken into account also the multiplicative factor ).
Comparing with experimental data of the fragmentation of asteroids
showed in Sect. 2, we can see that they are in better agreement (see
Table 4).
Name | Min P | Max P | S |
Prìbram | 37 | 110 | 50 |
Lost City | 6 | 18 | 50 |
Sumava | 0.6 | 1.7 | 1 |
Innisfree | 7 | 22 | 10 |
Space based obs. | 8 | 24 | 50 |
Space based obs. | 6 | 18 | 50 |
Benesov | 2 | 6 | 10 |
Peekskill | 4 | 12 | 30 |
Marshall Isl. | 60 | 180 | 200 |
Therefore, the new condition for fragmentation under unsteady regime is:
If we now apply the condition given from Eq. (15), taking
into account
,
kg/m^{3},
and that the scale height H=6.8 km at 30 km height, we obtain the
values in Table 5.
S [MPa] | ||
1 | 6.6 | 3.8 |
10 | 20.8 | 12.0 |
50 | 46.5 | 26.9 |
200 | 93.1 | 53.7 |
We have three reasonable solutions, which are underlined. If we consider that the final airburst occurred at about 30 km height, typical for type I bodies (see the discussion at the end of Sect. 7), we can consider 26.9 kms^{-1} as the most probable speed. This value is in agreement with first estimation from eyewitnesses (Cevolani et al. 1993).
For the Tunguska event, a more detailed analysis is in preparation in collaboration with the members of the Tunguska99 Scientific Expedition. Here we want to underline only one thing: from Fig. 1 we can see that, in subartic summer, the temperature does not change in the height interval crossing the troposphere. Therefore, the Mach number does not change and we can apply the Eq. (12), for steady state conditions, as shown in the Paper I. The error, noted by Bronshten (2000), introduced in considering the explosion height, instead of the fragmentation height is negligible: the new value is 16 kms^{-1} to be compared with the old one of 16.5 kms^{-1}.
Here we showed more details, both explaining better some assumptions in the theory of steady state motion outlined in the Paper I, and extending the theory to unsteady motion. We have taken into account the effect of turbulence and its interaction with shock waves. Some examples are discussed and we have found a reasonable agreement with available experimental data.
We have found two conditions for the fragmentation, according to steadiness of the motion.
Acknowledgements
I wish to thank R. Guzzi for providing the computer-ready files with data of US Standard Atmosphere 1976, and G. Longo for useful discussion. This research has made use of NASA's Astrophysics Data System Abstract Service. This work was partially supported by MURST cofinanziamento 2000.