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.
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.
Modeling:
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) :
ìï0.45( M ¥) + 0.424;
0 £ M ¥ < 0.722 üï
Eq. 1 Cd s ( M ¥) = í
ý
(-1.2( M ¥ + 0.35))
(- 2.2(M ¥ + 0.35))
- 8.9e
+ 0.92; 0.722 £ M¥ £ ¥ ïþ
ïî2.1e
2
ìï0.60( M ¥) +1.04;
0 £ M ¥ < 1.15 üï
ý
Eq. 2 Cd c( M ¥) = í ( -1.16( M¥ +0.35))
(-2.3(M ¥ + 0.35))
- 6.5e
+ 1.67;
1.15 £ M¥ £ ¥ ïþ
ïî 2e
2
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.
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].
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.
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.
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.