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Astron. Astrophys. 342, L1-L4 (1999)

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4. The importance of the stagnation temperature

Let us now consider the heating due to the conversion of kinetic energy of the flow into thermal energy, when the gas is brought to rest (in the boundary layer). This process can be described in terms of a steady flow energy in an adiabatic process:

[EQUATION]

where [FORMULA] and [FORMULA] are the stagnation and free stream enthalpy of the fluid, respectively, and [FORMULA] is the free stream speed. Note that the choice of the reference frame is not important here: if we consider a frame centered on the body, the fluid will move, and vice versa ; therefore [FORMULA] can be interpreted as the body's speed with respect to the atmosphere. We can rewrite Eq. (2) in terms of temperature:

[EQUATION]

where [FORMULA] is the specific heat at constant pressure. During the atmospheric path, as the Mach number is large, the meteoroid's speed is close to the maximum value corresponding to the stagnation temperature. Changes in the stream properties are mainly due to changes in the stagnation temperature [FORMULA], which is a direct measure of the amount of heat transfer.

This argument stresses the importance of the stagnation temperature in hypersonic flow, since it is related to the maximum speed of the stream, which in turn is close to the speed of the cosmic body. According to Shapiro (1954), the relationship between stagnation temperature and maximum speed of the stream can be expressed in the following way:

[EQUATION]

where [FORMULA] is the ratio of specific heats. By means of the equation of state for the air, [FORMULA] can be expressed as a function of the stagnation pressure and density:

[EQUATION]

In order to obtain a condition for the meteoroid breakup, the stagnation pressure [FORMULA] must be set equal to the mechanical strength S of the body. As for the stagnation density, we have [FORMULA] (Landau & Lifshitz 1987), where [FORMULA] is the undisturbed air density at the airburst height. Finally, by expressing [FORMULA] as a function of atmospheric height h and [FORMULA], like in Eq. (1), we obtain a new equation to estimate [FORMULA], which is close to the speed of the cosmic body at breakup V:

[EQUATION]

For [FORMULA] we can use a value of about 1.7, resulting from experimental studies on plasma developed in hypervelocity impacts (Kadono & Fujiwara 1996). Comparing Eq. (6) to Eq. (1), we see an additional factor of about 1.6. This comes from the fact that Eq. (6) derives from Eq. (4), according to which the stagnation temperature depends on speed when a body is travelling at hypersonic velocity. Eq. (6) shows that the airburst occurs thanks to the combined thermal and mechanical effects acting on the meteoroid. In other words, thermodynamic processes decrease the effective pressure crushing the body in a significant way, so the same body can reach a lower altitude, or for a given airburst altitude a lower strength is required.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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