360 
of the heat transfer to the surface is given by \, then 
the rate of mass loss will be: 
dm 
a, Vd dma 
Gh 7 De 
The problem still remains to determine the mass m of 
the meteoroid. Equation (2) will lead to such a solution 
when velocity, deceleration, density shape-factor (A) 
and air density are known. But to determine the air 
density from meteor observations, we must evaluate m 
independently. This can be done by equation (3) if we 
assume that the luminosity of the meteor is propor- 
tional to the kinetic energy of the mass lost and if we 
ean evaluate the proportionality factor. The greatest 
success in this evaluation has been attained by Opik. 
For photographic meteors of velocity greater than about 
20 km sec; Opik’s calculations [69] are reasonably 
well approximated by the relation: 
1 2 dm 
= Ti tSiVere= 
I= 7 Gv 
= * Am’ AVey (3) 
(4) 
where J is the intensity of the visual radiation and 7 V 
is the luminous efficiency, 7 being a constant. 
The rate of mass loss, from equation (4), becomes: 
dm 2r 
dea ave: (5) 
The mass at any instant f) is then obtained from the 
integral: 
ie 2 | ” C/V") dt. (6) 
From a completely observed meteor trail in which the 
velocity Vo and deceleration —V% are well determined 
at time f the air density pp can be determined by equa- 
tions (6) and (2) in the form: 
1/3 77! 
2° Vo 
eo) 1/3 
aj eee T/V* a| ; 
yAra V” U. at 
This form of the solution for p has been used by 
Jacchia [34], [in equation (10), his K = 21/8/(yA7)/*)] 
in recent Harvard results in the Naval Bureau of 
Ordnance program, as contrasted to the writer’s direct 
use [90] of equations (8) with (5) and (2) for the less 
numerous earlier meteor observations. In practice the 
numerical constants y, A and 7 of equation (7) may be 
evaluated separately from theoretical considerations, or 
as a single constant by comparison with atmospheric 
densities at the 50- to 60-km level where the densities 
are fairly well known. The agreement of the two 
methods is surprisingly good when Opik’s luminosity 
factor is used, y is taken as near unity, and a reasonable 
value of A is chosen. 
Jacchia [35] confirms Opik’s calculation of a decrease 
in 7 for velocities below about 20 km sec by studies 
of bright slow meteors in the 40- to 50-km zone. At 
greater altitudes an attempt [34] to minimize the rela- 
tive residuals in log p for the various meteors by varying 
the exponent of velocity in equation (7) indicates that 
(7) 
po = 
THE UPPER ATMOSPHERE 
a little improvement in the results can be obtained if 
the exponent of V in the denominator is reduced by 
0.5 to 0.8. 
We must conclude generally from Jacchia’s work 
that the largest systematic error (excepting a constant 
multiplying factor) in atmospheric densities as calcu- 
lated from meteoric decelerations from equation (7) 
rests in the assumed constancy of y with respect to 
both velocity and density. At extremely great heights 
(> 100 km) where we deal with fast, small meteoroids 
the value of the drag coefficient may possibly im some 
cases exceed 2 (7.e., y > 1). Where mean free paths are 
relatively large the re-emitted air material will mcrease 
the drag, as shown by Tsien [82] and Heineman [24]. 
Furthermore, the vaporized meteoric material will cause 
aN even greater increase by a similar process. As the 
height decreases, meteors of similar luminosity repre- 
sent larger masses moving at lower velocities in air of 
greater density. Hence one should expect the drag co- 
efficient to decrease. Consequently, the calculated 
values of air density from equation (7) may be some- 
what overestimated at great heights if correctly evalu- 
ated at lesser heights. This drag problem is under 
investigation. 
Less precise measures of atmospheric density can be 
determined from the luminosities early in the trails, 
while atmospheric pressure can be determined at the 
end points. Jacchia [86] finds, as did the writer [90], 
that the results obtained by these other methods are 
consistent with the more accurate results from the 
deceleration data when analyzed by the simple theory. 
Also, the light curves are accurately predicted except 
for flares. From the theoretical viewpoint it is interest- 
ing to note that the other methods involve \/f, and 
that the ratio \/(y¢) can be found by intercomparison 
of results. Jacchia [35] shows that this ratio requires no 
correction in the velocity exponent, suggesting that X, 
the heat transfer coefficient, and y, the drag coefficient, 
are truly constants or else show a similar velocity 
dependence. This conclusion is not surprising, at least 
from a qualitative point of view, but complete theoreti- 
cal determinations of \ and y are essential to an en- 
tirely satisfactory theory of the meteoric process. 
Atmospheric Density Distribution and Seasonal Vari- 
ations. As shown in the preceding section, the major 
uncertainty in the determination of atmospheric densi- 
ties by photographic meteor observations rests in a 
constant multiplymg factor. The density curve for 
meteors is, therefore, anchored to that obtained by 
other means in the 50- to 60-km zone. It is found that 
the gradient follows closely the gradient of the Tenta- 
tive Standard Atmosphere of the National Advisory 
Committee for Aeronautics [84]. This result is not sur- 
prising in view of the fact that the N.A.C.A. atmosphere 
at great heights was based, to a considerable extent, 
upon results obtained by meteor photography. 
The atmospheric densities found from the decelera- 
tions of thirty-five well-observed photographic meteors 
were compared by Whipple, Jacchia, and Kopal [93] 
with the densities of the N.A.C.A. atmosphere. The 
residuals show a strong correlation with season. In 
