A New Method to Constrain the Drag Coefficients of Meteors in Dark Flight.

R. Carter* (U. Oregon), P.S. Jandir* (U.C. Davis), M.E. Kress (Dept. of Physics and Astronomy, San Jose State University).

*Intern at NASA Ames through the NSF’s Research Experience for Undergraduates

Introduction:

Obtaining accurate values for the drag coefficients of meteorites is essential in order to determine their trajectory through the atmosphere. Some models that describe the descent of meteors use a constant drag coefficient. Others calculate the drag coefficient as a function of Mach number. However, experimental data suggest that drag coefficient depends on shape as well as Mach number. Due to the large number of possible shapes and orientations, it is unlikely that a single drag coefficient function can anticipate every possible landing site. To remedy this, we propose a method to bound the possible drag coefficients of a meteorite using two functions, one corresponding to the high drag extreme, and the other corresponding to the low drag extreme.

The best fit functions we created are a composition of two functions. For most of the subsonic regime, Cd is calculated using a quadratic function. At a critical Mach number, Cd is calculated by a sum of two exponentials. The best fit functions for drag coefficient as a function of Mach number in piecewise form for a sphere (Cds) and a cube (Cdc) :

Eq. 1

Eq. 2

Drag coefficients dependent on Mach number have been utilized in the past with positive results. The method proposed by Ceplecha et. al. in 1987 [1] was used successfully on a number of meteorites, namely the Bunburra Rockhole fireball [2], and the Neuschwanstein Meteorite [3]. We propose an alternative method of calculating drag coefficients as a function of Mach number, and introduce upper and lower drag coefficient functions that can be used to bound the landing ellipse of a meteorite.

Results:

The method introduced successfully by Ceplecha et. al. [1] lies between Eq. 1 and Eq. 2. Many previously used drag coefficient constants from 0.7 to 1.6 are also contained in the interval. In addition, Equations 1 and 2 agree with data from aeronautics, and conform to the expected behaviour of spheres and cubes. This suggests that Eq. 1 and 2 correctly bound the extremes of the drag coefficient.

Modeling:

We assume the upper limit of drag to be a cubic meteor, and the lower limit to be a spherical meteor. As shown in Figure 1, drag coefficient values change drastically between spheres and cubes, and between Mach 0 and Mach 5 for each shape.

Figure 2. This graph is a comparison of the proposed best fit functions, Eq. 1 (blue), Eq. 2 (red), the average of the two previous functions (green), and the drag coefficient function proposed by Ceplecha et. al [1].

Figure 1. A plot of the results of four experiments designed to find the drag coefficient of spheres and cubes at different Mach numbers. The red and blue pints are from experiment done to determine the drag coefficients of Spheres [3] [4]. The black and magenta points are from a similar experiment done on cubes, and a simulation [5].

Applications and Future Work:

Equations 1 and 2 can be used in computer models of meteorite descent to bound the size of a meteorites probable landing area. This is particularly useful in fireball networks when a meteorite enters dark flight. Typically, a meteorites probable landing area must be computed from its last visible trajectory by a fireball network. The drag coefficient plays an important role in this calculation.

To readily incorporate this Mach-dependent drag effect into our meteor model, we created best fit functions of the available data. One of these functions approximates the drag coefficient of a sphere at any given Mach number, while the other does the same for cubes.

If spheres and cubes are taken to be the aerodynamic extremes for a meteorite, then it can be inferred that a meteorites true drag coefficient at any Mach number will lie between the two functions.

REFERENCES:

[1]. Ceplecha, Z. (1987) Bull. Astron. Inst. Czechosl. Vol 32., 222

[2]. Charters, A. C., and Thomas, R. N. (1945) Journal of the Aeronautical Sciences, Vol. 12, 468-467.

[3]. Charters, A. C., and Thomas, R. N. (1945) Journal of the Aeronautical Sciences, Vol. 12, 468-467.

.[4].Hodges, A. J. (1957) Journal of the Aeronautical Sciences, Vol. 24, 755-758.

[5]. Anderson, J. D. (2006) Hypersonic and High Temperature Gas Dynamics Second Edition. American Institute of Aeronautics and Astronautics Inc., Reston Virginia.

Acknowledgements: MEK’s research program is supported by the NASA Astrobiology Institute via the Virtual Planetary Laboratory. PSJ and RC were awarded summer internships through the NSF’s Research Experience for undergraduates. We would also like to thank our friends from NASA Ames: Gary Allen, Jeff Brown, Stuart Rogers, Jeff Hollingsworth, Laura Iraci, and Prestin Martin. Thanks also goes to our advisors and teachers: James Schombert, Wendy Sullivan, Alejandro Garcia, Michael Kaufman, Steve Kevan, and Joshua Johnson.