Science in China Series G: Physics, Mechanics & Astronomy
© 2009
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How to estimate the effect of an intense meteor
shower on human space activities
WU GuangJie
National Astronomical Observatories/Yunnan Observatory, Chinese Academy of Sciences, Kunming 650011, China
In the present age, the potential threat to space projects coming from some intense meteor storms has
been noticed. Especially, the increasing activities of mankind in space for scientific, commercial and
military purposes have led to an increase in safety-related problems about the satellites, space stations
and astronauts. Several new techniques for observing meteors and meteor showers have been developed. However, how to estimate even predict the effect of an intense meteor shower should be further
studied. The initial definition about a meteor storm based on visual observations with a Zenithal Hourly
Rate of over one thousand seems insufficient, since it only means a storm or burst of meteors in
numbers. In 2006 the author suggested a synthetical index of the potential threats about intense activities of meteors; however, it is too complex to determine several parameters. In this paper, the author
suggests a Special True Number Flux Density (STNFD). Set a certain energy-limit, or a certain electric-charge-limit, and then calculate the number flux density. Through the comparison between two of
the 10 strong meteor showers in recent years it is found that the important factor affecting the space
flight security is not only the number of meteoroids, but also their velocities, their average energy and
the population index r. Calculations show that Giacobinids, even June Bootids, should be one of the
most hazardous meteor showers.
meteors, meteorites and tektites, spacecraft/atmosphere interactions, observation and data reduction techniques
The chance of a comet or other small celestial bodies
impacting the Earth is uncommon. However, their various debris: The meteoroids have been impacting the
Earth incessantly. In general, the meteoroids have not so
big or heavy as the man-made space debris, but they
have high velocities up to 11－72 km/s and high energies. In addition, the meteoroids can exist anywhere and
a tremendous number of meteoroids might be encoun－
tered in a short time[1 4]. Therefore, the threat coming
from the meteoroids may not be smaller than that coming from the man-made space debris. Moreover, the meteoroids may cause extensive destruction. Due to impact,
the shock waves can be generated and propagate along
the impacted bodies, compressing and heating both the
target and meteoroids themselves. A plasma cloud may
enclose the target and expands into the surrounding
vacuum, emitting electromagnetic radiation in a widely
spectral range.
In fact, the increasing activities of mankind in space
for scientific, commercial, and military purposes have
led to an increase in safety-related problems. The actual
destruction of space projects has been recorded and reported, like the International Sun Earth Explorer
(ISEE-1) which lost its 25% of the data in 1977 and the
Olympus Telecommunication Satellite of ESA which
lost its control and culminated in an early end of its mis－
sion in 1993[5 9].
In general, Leonids is regarded as one of the most dangerous meteor streams, even though under the normal
conditions, some people thought that the level should
－
have been closer to 15000 in 1966 than 150000[8 10].
Received July 23, 2008; accepted February 20, 2009
doi: 10.1007/s11433-009-0167-1
†
Corresponding author (email: [email protected])
Sci China Ser G-Phys Mech Astron | Aug. 2009 | vol. 52 | no. 8 | 1161-1168
However, based on their astronautic definition of the
meteor storms, Ma et al. thought the Giacobinids (Draconids) in 1998 might be the strongest one among the 10
active meteor showers in recent years[11]. Anyway, the
initial definition about a meteor storm coming from a
Zenithal Hourly Rate (ZHR) of above one thousand is
not sufficient. As for the problems how to define and
forecast the intense activity of a meteor storm and how
to reduce the underlying damage as far as possible, dissectional and groping work is still needed.
In the last paper, a synthetical index denoting the activity and potential threat of an intense meteor shower is
suggested[12]. However, the parameters in the synthetical
index should be further analyzed and discussed. In this
paper, these important factors affecting the space flight
security have been pointed based on the calculations of
the 10 active meteor showers. In addition, according to
the newly defined Special True Number Flux Density, it
is shown that among the most hazardous meteor streams
there should be Giacobinids, even June Bootids.
where NObs is the total number of meteors recorded by
the observer, Teff the effective observing time in hours
and hr the elevation of the shower’s radiant at the observing time. The perception correction cp varies between 0.4 and 2.5, having a median value close to 1.0[14].
Usually, we can adopt γ = 1.0[15,16], but in some cases its
value may be up to 1.4, even 1.8[14,17]. These measures
may be obtained by different observers at different
places in variable environments, however, through the
above formula they are reduced under a standard condi－
tion and can be compared worldwide[14,16,18 21].
For the whole visual observations the ratio of the total
true number to the observed one N Obs = ∑ N m,Obs is
1 The Zenithal Hourly Rate and ZHR*
Though several new kinds of techniques for observing
meteors have been developed in modern meteor astronomy, the visual observation is still the most popular
and most fundamental. From the simple records of the
number and magnitudes of the meteors, the ZHR and the
population index r of the meteors can be calculated and
compared worldwide, indicating the intensity of the meteors and their magnitude distribution, respectively[13].
The population index r represents the ratio of the true
numbers of the meteors in two consecutive magnitude
classes of (m) and (m−1):
N
r= m .
(1)
N m −1
In general, it is assumed that r is a constant at least in
the whole magnitude range covered by the visual observations. This can give a simple relation:
N m = N0 × r m .
“perception function” and Δm is the magnitude difference:
Δm = LM − m,
(4)
where LM is the limiting stellar magnitude for this observation.
The observed ZHR is defined as
N
ZHRO = Obs × r 6.5− LM × sin(hr )−γ × cp−1 ,
(5)
Teff
(2)
+∞
c(r ) =
∑ r −Δm
∑ Nm =
0
,
+∞
∑ N m,Obs ∑ r −Δm × P(Δm)
(6)
0
which can be used to get the total true number of meteors through the corrections of NObs or ZHRO.
In recent years, we suggested an analytic function:
P(Δm) = 0.5 + 0.505 × tanh(0.66Δm − 0.013Δm 2 − 2.43),
(7)
while Δm < 9.4633, otherwise P (Δm) ≡ 1, to calculate the probability of perception P (Δm). In addition,
eqs. (6) and (7) are used to calculate the values of the
c(r ) directly[13,22].
For studying the potential hazard of a strong meteor
shower, Ma et al.[11] suggested a special Zenithal Hourly
Rate ZHR*:
ZHR∗ = ZHRO × 1, 000 ×
c( r ) ⎛ M 6 ⎞
×
Ared ⎜⎝ M χ ⎟⎠
s −1
However, in visual observation, since a meteor moves
fast and stays in the sky for less than one second normally, only a fraction of meteors can be caught by a naked eye. The number of the observed meteors at a certain magnitude m is
(3)
N m,Obs = P(Δm) × N m ,
where M6 and Mχ are the masses of the meteoroids with
Magnitude 6 and Magnitude χ, respectively. s is the
mass distribution index[20]:
s = 1 + 2.3 × log(r ).
(9)
where P(Δm) is called the “probability of perception” or
Ared is called the reduced area, which represents the
1162
WU GuangJie Sci China Ser G-Phys Mech Astron | Aug. 2009 | vol. 52 | no. 8 | 1161-1168
,
(8)
area surveyed by an observer at the meteor level (the
height H = 100 km)[19,20]:
Ared = 37200 × (r − 1.3)
−0.748
2
km .
(10)
In fact, the ZHR* is defined as the number of meteoroids passing over the zenith area of 1000 km2 per hour
and each meteoroid can form a crater which is not less
than 1 cm in diameter on an aluminum surface. It means
M χ V02
2π 3
=
(11)
R ρε ,
2
3
where R is the radius of the least crater impacted, ρ the
mass density of the spaceflight instrument and ε the requested energy to destroy 1 g material. The geocentric
velocity V0 is in units of km/s and can be got from its
velocity V∞ in space:
V02
[11]
= V∞2
+ 2 × 398600.5 / 6470.
(12)
*
Ma et al. thought the number of ZHR should be directly proportional to the impact probability, so that,
they call it “the astronautic definition of a meteor storm”.
Performing the calculations, Ma et al. gave the result
that 1998 Giacobinids has had a ZHR* as high as 8429[11],
much greater than the others among the 10 active meteor
showers since the 1990s.
2 The ZHR* and True Number Flux Density
ten as
ZHR∗ = ZHRO × 1,000 ⋅
c(r ) mlow −6.5
.
⋅r
Ared
(17)
From eqs. (17) and (15), one may obviously have
ZHR∗ = 1000 × QN ( E ≥ Elow ).
(18)
Here, we rewrite the range of m ≤ mlow to E ≥ Elow ,
since Ma et al.[11] set the Elow directly, but not mlow.
The above equation says that the ZHR*, in nature, is a
kind of number flux density.
3 The relationship between the mass
and the luminosity
For determining the mlow from Elow, you need know
about their relationship. In the literature several kinds of
empirical mass-magnitude formulae for meteoroids have
－
been used[18 20,23,24]. Among them, we prefer to use
m = 14.247 − 2.5log M − 10.0log V∞ mag.,
(19)
[12,25]
where the mass M is in units of gram
formula, it is easy to get
M
m1 − m2 = 2.5 × log 2 .
M1
. From this
(20)
According to the definition of the magnitude, we have
L
m1 − m2 = 2.5 × log 2 ,
(21)
L1
In the study of the meteors, the True Number Flux Density of the meteors with a magnitude of mlow at least can
be reduced as[12,19,20]
∑ Nm .
QN (m ≤ mlow ) =
(13)
T ⋅A
where L1 and L2 are the luminosities of the two meteors.
Therefore, we can get the simplest “ratio of the mass to
luminosity” as
L = C ⋅M,
(22)
For mlow = 6.5 it can be written as
where C is a constant. If we adopt another formula,
like[19]
eff
QN 6 = ZHRO ×
red
c(r )
,
Ared
(14)
L = C ⋅ M b,
QN (m ≤ mlow ) = QN 6 ⋅ r mlow −6.5
(24)
where b = 0.92, which also appears in the relationship:
(15)
On the other hand, since the ratio of the cumulative
numbers of meteoroids brighter than M χ and M 6
can be written as[19,20]
φ ( M χ ) ⎛ M 6 ⎞ s −1 mlow −6.5
,
=
=r
φ ( M 6 ) ⎜⎝ M χ ⎟⎠
(23)
the “ratio of the mass to luminosity” will become
so
ZHRO × c(r ) mlow −6.5
=
⋅r
.
Ared
m = 40 − 2.5 × log(2.732 × 1010 M 0.92V03.91 ),
α X = 7.7 × 10−10 M 0.92V03.91 m−1,
where α X is the maximum electron line density along
the meteor trail[26]. In like manner, Hughes et al. gave a
succinct formula:
α X = 2.0 × 1010 MVi 4 m−1,
(16)
then, the ZHR* introduced by Ma et al.[11] can be rewrit-
(25)
(26)
where the mass M in units of gram and the impact velocity Vi of the meteoroid with the atmosphere of the
Earth is in units of km/s[27,25,21]. It is obvious that the
WU GuangJie Sci China Ser G-Phys Mech Astron | Aug. 2009 | vol. 52 | no. 8 | 1161-1168
1163
exponent of b = 0.92 is close but not equal to a unit. In
addition, the exponent of 3.91 in eq. (25) is close but not
equal to 4. These exponential non-integers will make
other calculations more complex, but do not change the
result largely.
4 The definition of a special true number
flux density
As a special number flux density, the definition of ZHR*
gives us a good ideal. Therefore, we can define a new
Special True Number Flux Density (STNFD):
QN∗ = ZHRO × β1 × β 2 ,
(27)
where
β1 = 10000 ×
c(r ) −6.5
⋅r ,
Ared
(28)
β 2 = r mlow .
(29)
The calculation range can be set as m ≤ mlow ,
E ≥ Elow or α X ≥ α X ,low and so on. The
QN∗
is the
number of meteoroids passing over the zenith area of
100×100 km2 (about the order of the central area surveyed by an observer) per hour and each meteoroid has
an mlow , Elow or α X ,low etc., at least. Obviously, it
should be directly proportional to the impact probability.
Tables 1 and 2 directly show you the effect of r and
mlow to the values of β1 and β 2 . Table 1 indicates
that except for r = 1.4, which is a little special, the larger
is r, the smaller is β1. However, the value of β1
keeps in a small range with a difference about one order,
while β 2 has a very large variation with r. In addition,
Table 2
r
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
1164
when mlow is larger or smaller than zero-magnitude, β2
has an inverse variation: that is very important!
Table 1 The variation of c(r) in eq. (6), Ared in eq. (10) and the parameter
β1 in eq. (28)
r
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
c(r)
3.43
5.22
7.33
9.69
12.23
14.91
17.67
20.47
23.27
26.05
28.79
31.49
34.12
36.68
Ared
208229.8
91549.6
62476.0
48574.6
40250.3
34640.3
30571.3
27468.0
25013.1
23016.3
21356.1
19951.7
18744.9
17696.3
β1
0.01849
0.02687
0.02570
0.02203
0.01807
0.01454
0.01160
0.00924
0.00737
0.00589
0.00473
0.00382
0.00310
0.00253
The value of m low can be changed into Elow and
V∞ , since
mlow = 14.247 − 10 log V∞
2 Elow
⎛
⎞
− 2.5log ⎜ 2
⎟
⎝ V∞ + 2 × 398600.5 / 6470 ⎠
= 13.4944 − 2.5log Elow − 10 log V∞
+ 2.5log(V∞2 + 2 × 398600.5 / 6470).
(30)
Table 3 gives the calculated values. Since (2×
398600.5/6470) = 123.2, which is much smaller than
V∞2 , in general. Therefore,
mlow ≈ 13.4944 − 2.5log Elow − 5.0log V∞ ,
(31)
is their main relationship.
The variation of the parameter β2 in eq. (29)
mlow = 8.0 mag.
14.757891
42.949673
110.199606
256.000000
548.758735
1100.753142
2088.270646
3778.019983
6561.000000
10995.116278
17857.939049
28211.099075
43477.921385
65536.000000
mlow = 6.0 mag.
7.529536
16.777216
34.012224
64.000000
113.379904
191.102976
308.915776
481.890304
729.000000
1073.741824
1544.804416
2176.782336
3010.936384
4096.000000
mlow = 4.0 mag.
3.8416
6.5536
10.4976
16.0000
23.4256
33.1776
45.6976
61.4656
81.0000
104.8576
133.6336
167.9616
208.5136
256.0000
mlow = 2.0 mag.
1.960
2.560
3.240
4.000
4.840
5.760
6.760
7.840
9.000
10.240
11.560
12.960
14.440
16.000
mlow = 0.0 mag.
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
mlow = −2.0 mag.
0.510204
0.390625
0.308642
0.250000
0.206612
0.173611
0.147929
0.127551
0.111111
0.097656
0.086505
0.077160
0.069252
0.062500
WU GuangJie Sci China Ser G-Phys Mech Astron | Aug. 2009 | vol. 52 | no. 8 | 1161-1168
mlow = −4.0 mag.
0.260308
0.152588
0.095260
0.062500
0.042688
0.030141
0.021883
0.016269
0.012346
0.009537
0.007483
0.005954
0.004796
0.003906
The value of mlow is determined by Elow (unit: J) and V∞(unit: km/s)
Table 3
Elow
70.7×10−2
70.7×10−1
70.7
70.7×101
70.7×102
70.7×103
70.7×104
70.7×105
70.7×106
V∞= 10.0
V∞= 20.0
12.24
9.74
7.24
4.74
2.24
−0.26
−2.76
−5.26
−7.76
10.16
7.66
5.16
2.66
0.16
−2.34
−4.84
−7.34
−9.84
V∞= 30.0
V∞= 40.0
V∞= 50.0
V∞= 60.0
V∞= 70.0
9.12
6.62
4.12
1.62
−0.88
−3.38
−5.88
−8.38
−10.88
8.44
5.94
3.44
0.94
−1.56
−4.06
−6.56
−9.06
−11.56
7.93
5.43
2.93
0.43
−2.07
−4.57
−7.07
−9.57
−12.07
7.52
5.02
2.52
0.02
−2.48
−4.98
−7.48
−9.98
−12.48
7.17
4.67
2.17
−0.33
−2.83
−5.33
−7.83
−10.33
−12.83
Obviously, both values of β1 and β 2 are determined by r, V∞ and Elow . The lowest catastrophic
energy Elow can be set by the special research, however, the factors of r and V∞ are the inherent characters
of a meteoric shower.
5 The efficiency of the transition of energy
We see that the question about the astronautic definition
of a meteor storm suggested by Ma et al.[11], in substance,
is a kind of STNFD. The only special matter is the critical energy Elow being set as[11]
πρε
= 70.7 J.
(32)
Elow =
12
They hope that every meteoroid must have the ability to
create a crater no less than 1 cm in diameter on an aluminum surface, but they might set the critical energy too
low.
The question is whether or not all of kinetic energies
of the meteoroid can be changed into the heat creating
the crater without any waste.
For the impact of Leonids meteors with the Moon, a
value of s = 1.6 ± 0.1 might be appropriate, which corresponds to an r of 1.82[28]. The flashes are mainly the
result of thermal emission from hot plasma plumes created by the impact[28,29]. An important parameter in the
impact of the meteoroid with the Moon is the fraction of
the initial kinetic energy converted into the radiation or
luminous efficiency η. From the observations of 1999
Leonids impacting the Moon, a value of η = 2 × 10
−3
in
the wavelength range of 400－900 nm was obtained
with an uncertainty roughly one order of magnitude.
This efficiency is significantly larger than that previously estimated, since very high-velocity collisions like
Leonids are much more difficult to reproduce in the
laboratory.
Certainly, the transition efficiency η must also exist in
the impact of the meteoroid with the spaceflight instrument, which might be as large as 2×10−3 or less.
As an example, we can still use the 10 active meteor
showers as Ma et al.[11] used in their paper. Through eq.
(30) and by adopting Elow,1 = 70.7 J, mlow,1 can be obtained. Our calculations are listed in Table 4. It is surprising that the mlow of the showers 1998 June Bootids
and 1998 Giacobinids can be as faint as about 10.0 mag.,
which is much lower than the visual limit and makes the
QN∗ be very large. By considering the transition efficiency, setting Elow,2 = Elow,1 /(2 × 10−3 ) = 3.535 × 104 J,
even considering that the efficiency η may be lower
again about one order and setting Elow,3 = Elow,1 /
(2 × 10−4 ) = 3.535 × 105 J, the calculated values are also
given as QN∗ ,2 and QN∗ ,3 in Table 4.
It is obvious that even η = 2 × 10−4 , 1998 Giacobinids and 1998 June Bootids still have a positive magnitude of mlow,3, that is because they have lower velocity
of 20, even 18 km/s. Moreover, r = 3.0 is the largest
value among the 10 showers, which made 1998 Giacobinids has a larger β 2 value. In general, 1998 Giacobinids has a mlow about 3 mag. fainter than that of
Leonids, and about 2.6 mag. fainter than that of Perseids.
Therefore, 1998 Giacobinids has the biggest QN∗ ,2 and
QN∗ ,3 and is one of the most hazardous meteor showers.
Certainly, if on the Moon, Perseids may be one of the
most hazardous meteor showers[12]. However, anyway,
do not ignore June Bootids, which always has the faintest mlow and may have a larger threat either on the Earth
or the Moon.
In Table 4, the value of r is supposed as a constant for
a meteor shower. Actually, people thought that r can be
WU GuangJie Sci China Ser G-Phys Mech Astron | Aug. 2009 | vol. 52 | no. 8 | 1161-1168
1165
Table 4 Calculations of 10 meteor showers, by setting Elow,1 = 70.7 J, Elow,2 = 3.535×104 J and Elow,3 = 3.535×105 J
Meteor shower
V∞
1991 Perseids
1992 Perseids
1993 Perseids
1994 Perseids
1995 α Mon
1998 June Bootids
1998 Giacobinids
1999 Leonids
2001 Leonids
2002 Leonids
59
59
59
59
65
18
20
71
71
71
r
1.9
2.1
1.8
1.8
2.51
2.22
3.0
2.3
2.25
2.92
ZHRO
mlow,1
mlow,2
QN∗ ,2
mlow,3
QN∗ ,3
350
250
300
250
350
102
720
3700
3430
2940
7.55
7.55
7.55
7.55
7.34
10.4
10.2
7.14
7.14
7.14
0.81
0.81
0.81
0.81
0.59
3.70
3.41
0.39
0.39
0.39
14.09
9.11
12.39
10.32
7.74
34.43
224.66
83.37
80.88
36.13
−1.69
−1.69
−1.69
−1.69
−1.91
1.20
0.91
−2.11
−2.11
−2.11
2.83
1.43
2.85
2.38
0.78
4.69
14.41
10.39
10.65
2.48
supposed as a constant for a trail erupted in one return of
a comet. However, a meteoric storm may be a mixed
stream including several trails erupted in different returns. As an example, the value of r is indeed different
for the 4 showers of Perseids, is also different for the 3
showers of Leonids in Table 4. But then, we can still use
a constant for a meteor shower. Reason 1, r represents a
distribution of the numbers along magnitude according
their brightness or sizes of the particles in a stream. In
fact, r is not a strict constant in the whole visual magnitude range but approximately. Reason 2, the value of r is
dynamic at any moment. It means the distribution-self in
a stream is not uniform. Reason 3, in a mixed stream,
since all of the particles are erupted from one and the
same comet, in general, the value of r can be preserved
in a reasonable range, which may be much different
from other stream or comet, so that a new r could be
found to represent the mixed distribution approximately.
Reason 4, as a matter of fact, nobody find that a constant
r cannot be used even in a short-time meteor observation.
6 Discussion
[12]
In our last paper, Wu suggested a synthetical index of
the potential threats about intense activities of meteors.
In this index several aspects of the intensities are considered, such as the number, mass, impulse, energy and
electric charge, etc. For different special purposes, the
parameters can be modulated. However, for determining
these parameters correctly, a lot of work may be needed.
In fact, eq. (19) gives an important relationship about
the magnitude, mass and velocity. Rewrite this equation
as
m = 14.247 − 2.5 × log( MV∞4 ),
will be equal to MV∞4 . Therefore, the larger the velocity, the smaller the mass M, momentum MV∞ and energy MV∞2 / 2. Eq. (26) constitutes also a very important relationship between the mass, velocity and maximum electron line density. The comparison of eqs. (33)
with (26) says that a bright meteor must have the larger
maximum electron line density as
m ≈ 40.0 − 2.5 × log(α X ).
(34)
In other words, if the streams have the equal number
flux density QN in a certain magnitude range, the
stream with a larger velocity V∞ will have the smaller
mass flux density QM , smaller impulse flux density
QI , smaller energy flux density QE and about equal
charge flux density QC . In a word, the velocity V∞ is
a very important parameter and the multifarious flux
densities ( QN , QM , QI , QE , QC ) may have certain relationships.
See the tables, in Table 1, except for r = 1.4 having a
little difference, the parameter β1 becomes smaller
while r is larger in the range of r = 1.6 to 4.0. However,
the value of β1 changes only about 7.3 times. In Table
2, β 2 varies largely with the variation of r. Since
β 2 = r mlow , the variety is anti-symmetric. For mlow = 0,
the value of β 2 keeps a unit. For mlow > 0, the value
of β 2 becomes larger with the increase in r. For
mlow < 0, the value of β 2 becomes smaller with the
increase in r. For mlow = ±2, the value changes about
8.16 times in the range of r = 1.4 to 4.0. Since r 8 =
(33)
(r 4 ) 2 = ((r 2 )2 ) 2 , for mlow = ±8, the value will change
which means that the meteors with the same magnitude
about 4440 times. Therefore, in the conversion of
1166
WU GuangJie Sci China Ser G-Phys Mech Astron | Aug. 2009 | vol. 52 | no. 8 | 1161-1168
ZHRO to QN∗ in eq. (27), the effect of β 2 may be
obtain the QN∗ only.
larger than that of β1 , in general. Of course, the effect
then, the population index r for the natural characters of
meteor streams. The relationship of V∞ with QN ,
As you know, the Leonids meteor shower has been
known as the strongest meteor stream. Indeed, it has
－
been an outstanding exhibition in history[3,18,30 34]. For
the future, Yeomans et al. hold that because of planetary
perturbations, it will be another century after the 1998－
2001 events before the significant Leonids meteor appears once again. Moreover, the Leonids may have gone
forever after AD 2500[30].
It is a pity that in literature we have not found any
reports which confirm the endangerment of Giacobinids
or June Bootids.
For the future prediction, since scientists know the
velocities V∞ of the main meteor showers very well,
QM , QI , QE and QC tells us, in most cases, that only
the important question is to set a reasonable Elow (or
the consideration of QN may be enough. The defini-
M low , I low , Clow and so on), and the most possible
tion of QN∗ tells us that in different cases, we can set a
population index r is estimated.
special mass-limit, impulse-limit, energy-limit or
charge-limit, of course, and a magnitude-limit, and then
The authors would like to thank Mr. Ling Zongxu for the improvement on
English of this manuscript.
of β1 makes the QN∗ ZHRO , in general. See Table
3, when Elow is set and if V∞ is larger, then the
needed mlow must be brighter. In addition, the value of
mlow
changes faster along V∞ than along Elow .
While V∞ changes from 10 km/s to 70 km/s, it is
needed that the value of Elow changes over two orders.
All of the analysis mean that in the conversion of
ZHRO to QN∗ , the firstly important parameter is V∞ ,
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