A general formula which expresses the distance along the meteoric fireball trajectory 1 as a function of t is discussed. Differential equations which include the motion and ablation of a single nonfragmenting meteor body are presented. The importance of the atmospheric density profile in the meteor formula is emphasized

Ceplecha, Z.

Pecina, P.

English

NASA Technical Reports Server

f
I
f
N85-20473
ATNOSPIIERIC-PROFILE IMPRINT IN FIREBALL ABLATION-COEFFICIENT
Z. Ceplecha and P. Pecina
Astronomical Institute of the Czechoslovak Academy of Sciences
251 65 Ondrejov Observatory, Cztchoslovakia
€
During the past two decades three different projects for registration of
meteoric fireballs were put into operation using multistation photographic
technique. Thcy have yielded data on _everal hundreds of fireball trajectories,
some of them with deep atmospheric penetration down to t,eighta of 20 kilor_eters,
The itmediate results o£ multistation p_.otogr_phof a fireball ar'_the relative
distances along,.,the trajectory, loss.' and the heights, ho_s, measured at
each shutter trme-mark, t, (shutter breaks of the image)._ The precision of one
value of I _° is of the order of several tens of _eters. There are usually
many tens _°independent poiuts (breaks) available for long fireball
trajectories with independently m_-asured lobs and h_b.. We need a good
.. theoretical relation for a least-squares solution o_ _-1(t) or h=h(t), where l
is the _heoretieally given distance along the trajectory and h the height.
• Until recently there were no adequate formulae expressing theoretically the
distance along the fireball trajectory, I, as function of t. We have been able
,-'. to find such formulae and moreover to find their general form for any
atmospheric profile used.
The motion and ablation of a single non-frag_nentingmeteor body can be
expressed by the following set of dif_erentlal eqvations first presented by
IIOPPE (1937):
,/
dv /rOd-213 m-I/3 2
.. d-Y= - r o v (I)
dm ._.A -213 213 3
d-{'- 2-_ °d o m v (2)
dh (3)dt = - v cos zr
...
.... • where,v is the velocity, m the mass, h the hei_Lt of the meteoroid, _ the air
density at any tis_einst-_ntt. The parameters are: r the dra_._oefficient,
! h the heat-transfer coefficient, A the shape factor: A = S m-_13 #)213,
where S is the head cross-section and od tb.e density of the r_eteoro_d, t, is
the energy neceesary for ablation of one unit of r,nss and zp is the nngle
between :he fireball trsjectory and the vertlcal. The ablat'ion coef£icient_
is defined by
t,
o = 2€r (4)
and the _hane-dcnsity cocfficient, K, by
K = rh Od-2/3 (5)
To solve the system of equations (I) to (3), we have to nsstme a density profile
o£ the atmosphere° Until now everybody _,orl.ingin the field of physical theory
of r,eteors and its application to observations solely used the assumption of
exponential decrease of the air density with height corresponding to an iso- . .
: thermal atmosphere: i
_ = Vc exp(-bh) (6) i
https://ntrs.nasa.gov/search.jsp?R=19850012163 2020-03-22T18:36:59+00:00Z
L'
• .. ', . .
2_
I I 123
{ where b, the air densit) gradient, was assumed constant as well as th_ zero-
" level air-density, Pc" Horeover, the solution of the system (I) to (3) was
known only in the form of v=v(p):
2K_mZI13e*p(_uvz)
t -- 1 ov 2) ov2).= (7)
, El(6 - El(6 b cos zR
where v_ is the initial velocity (_efore entering the atmosphere) and m is the
• . . -" _ t*X )_ll)_U . . .
znltxal =ass and E=(x) = _cxlRT-- Is the exponential integral. Thus
[ " numerical differentiation of directly observed distances along the fireball
' trajectory, I . was necessary to get v __ for application of formula (7)
, . ODa m u_ ....
to observatxons. _uch an indirect metho_ yxelded ablatxon coeffxcxents and
I initial velocities with standard deviation_ much larger than corresponding to
the accuracy of the measured distance, lo. so the accuracy of the observed
i quantities was far from being utilized fu_y,
Recently we have proposed a solution of the system (1)-(3) in a closed fo_
;: expressing l=l(v(t)):i
" 2 /:o o2,d
t - t b cos z R 2 - 1 = 1 ($)
o X [El(_ OV 2) - El(_ ox2)
- 1 - 1 2
" | = 1 ln[Ei(_ ov 2) - El( _ ov /
o cos _- : I ] (9)
Eere v , l° are the velocity and the distance ; the point t =0 from
; i where _he relative time iS counted. The integration varxable xs denoted x. The• . o . t
P equations (8) and (9) hold under the assumption (6). The obeerved values,!
lob s and h - for each independent time instant, t, can be fitted to ;
equations _ and (9) by the least-square method and the parameters lo, We, ;
v , o can be determined.
i
: In applying our formulae (8) and (9) to observations of fireballb we
I. started to suspect _uch greater importnnce of the air density profile for the
, resulting values of. the parameters than it was as6t_ed previously, when the .-
simplistic approach to the air density profile with a constant air density
T gradient was assumed as correct. Our solution of (1) to (3) for a general air
density profile has the form
1
- to = /iv-ldi (1o)
o
- 1 - 1., El( _ ov= 2) - EI(_ ov 2) dh
= -_r----- 111)
- 1 - 1
El(6 °v'2) - El(6 °We2) fh 0 dh
o
where h is the height at the tie,e instant r . Amor,L other parametur_ v0 • . 0
and ,_ can be determined _y the least-square method to fit the observations to
(I0) and (II).
' Details on both solutions (8), (9) and (I0), (II) can be found in t,*o
' recent papers (PECINA and CEPLEQiA, 1983, 19_4) together t_ith outline_ of the
'. _ numerical procedures and of the computer progrums used. The ablation
y:
_L
/.
t: 124
coefficients and initial velocities €omputed for 10 Praitte l;etvork _nd e_e
Euto_c&n Hetwork ftreballs for the isotbes_al at_ostt_ere (iqO2) and fL,r tl,e
_" seasonal aleutian',ere (19bO) are €,3cparl_' in Table I Cral, htcal ¢onpartson o[! •
&blatton cuelticients cent:used for the iJutF, et-_al artiest'here and let the
teatc_l atmosphere are plotted tnt trots l. The lolls'wilt; Irsults are evident.
a) "lhe _o_;'uted _blation coefficient i_ str!,n;'ly del,_ndcnt _,n the at_esl,tcric
_o,t_l uJed. Pxllrr,.nces by ustnt_ a tte.plttttc isothermal ethos|here ate up
to factor el two.
b) The standard deviation8 vhrn ostn_ a seasonal atnos_l'ete are *itnificxntly
s_aller than lot tl,e .it.pie tso[helx_al t,odel.
€) _he xttittal voloctty dtfter_ also tar t, utttde the standard deviations let
the t_jority of caves and the values [ro_ the statue.st attest,here are
better. ]hit |_ts ast_on_ical ti_niltc.mce in ct3cs_uttnl_ the ort, tts.
The nain conclusion it rvi_tnt. The _rneroua ate_ption of si_pte
atuosl.heric _odel utcd up tu now for theoretical €onside:atit_ns of _eteoroid
i_enetration into the at,'ospherr and for computational ap;,lications to ftrel, alla
yields incorrect results. At least, the density I.ruf:le of "Eontl, ly
at=ospberes" should Le !iled (C]I_, 1972) l*,r _ny future tt_eoretical a:!_
expertnental applications to gel any reliable data on ablation coefftczet_tz and
; initial veloctttes of fireballs vith t_o_d dynamic data. Ar.alysin_ aUl*tt_n
coe[ficients conputed for [_ny ftleballt o_ different structure and co_t.osttion
cf their ccteorotdt, ue could better recognite differr_t tyl'e_ of _'_dtts and
:_teit avvrage charac_.eri|tl¢s. Then, usin_ the average statistical value of the
ablation coefl_cient for each separate _tructnral and €o_.l_sition.tl t_oup ,,_f
! fireS.alia, ve could determine detail* oI the instant axe drnetty I_toftle of the
* Hiddle AiDe,photo at the particular _e_ent of _y fireball wtth );ood _ynautc
i data and _oreover the local disturbance t,y large ueteuric bodies. XnJ in tl, it
directi_,n_e _ant to contribute to *tudies vf the ltiddleAt=o|vhere tn the near
i future.
¢IP,A (1972), OOSI'ARInternation_L_rf._ren(_,,_iro_rhor_ Io72_,_kadroie Vrrlal:.
_erlin.
_e .-__ecir.a, P. and .. CeI'lecl:a (1983), _tll. 8_tron. Inst. Czecl-,osl., 3", IU2.
Poring, P. and Z. _eplecba (196_,), _ull. Astron. last. ¢_.cc1_1., _, _o. l
(cr 1983. I;o.6).
U,S. Standard Atmosphere (1962), _a,hint:ton
U.S. St_adard At_osl, here St.pplt'_en_s (19bO), _amhin_ton
• - • ° . • , , . .
\125
Table 1. ¢c_pariJon of ablation coefficient, ,,, and initial
velocity0 v_. computed with constant air drnlity
KradienL, b, and wtth £eat_nal air density profile.
._ s'/kza" v L_/s
I llei ght
Firebal I ' Seasonal . Sessona I Int e_0al
Eu=ber . -,. exp(-bh) .Atmosphere IL. -,,.ez_'(-bh) AL_osphere Im
PH39057 0.O229 O.0195 14. ] 39 It,. 350 74
•.0009 ..0005 -".006 ..002 36
!'N39060 0.007 0.01B _.1.9_ 31.81 91
• ,003 _.002 t .00 _..04 50
I_39065 0.036_ 0.0316 17.303 17.332 68
• .0009 , .00C_5 :,00') ...00-_ 34
t_t3907B 0.0614 0.060._ 10.915 10.982 b2
...0031 ).0013 :.OIl -'.003 41
1"t139404 0.0195 I 0.0303 15,319 15.)46 79
• ,0007 _..0007 _..006 :.roY., 2E
PN3gt.05 0.0t.65 0.0t,51 It, ,t,OS 1_..385 70
_..0017 "-.0009 •,OfI ._.003 44
FN39434 0.0_ 9 0.0146 14.._89 14.317 69
• .0009 t.0007 "-.00£ -%002 27
PN]9._69A 0.0I32 0.0l76 26.39 26 ./,2 95
r :.0020 -'.0016 *.0_ .%02 55
P1139729C 0.0107 0.0109 27.$._0 27.823 72 '
- - ;.0004 -.0002 _,0.9 _*.009 43
P.;39$20A 0.0203 0.0212 24.617 24.611 E0
-'.0003 -'.0002 .*.006 t.O03 39
_290181 0.059 0.011 II.578 II.611 64
roO_ _'.004 _..Oil -+.004 33
• Different average values o( b were used €orresponding to different
height intervals for _articular fireballs atnd to the U.S. Standard
At=osphere (1962).
Seasonal at=omphere were taken from U.S, Standard Atmosphere
Supple_ents (1966).
• A
+ /
A! "!- 4"
Figure I. The average ablation coefficient, _,
,'_ co=pute_ frcu the ai=ple isothermal atmospheric
_odel (_' - .,, exp (-bh)) is plotted against
the _verage ablation coefficient computed froD
the sea6or_l atmospheric _odel. Bars are the
standard deviation,. The 45" line _rk8 equal
values of _' fro_ both con_,utations.
/


Atmospheric Profile Imprint in Firewall Ablation Coefficient

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server