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Nodal regression of the quadrantid meteor stream: An analytic approach
ICARUS 49, 125-134 (1982)
Nodal Regression of the Quadrantid Meteor Stream" An Analytic Approach C. D. MURRAY Center for Radiophysics and Space Research, Space Sciences Building, Cornell University, Ithaca, New York 14853 Received September 25, 1981; revised November 2, 1981 The mean orbit of the Quadrantid meteor stream has a high eccentricity and inclination with an aphelion close to the orbit of Jupiter. The nodal regression rate, a quantity which has been well determined from observations, cannot be calculated with sufficient accuracy using standard loworder expansions of the disturbing function. By using a high-order expansion of the disturbing function we show how the behavior of the longitude of ascending node of the Quadrantid stream is a result of both secular and resonant effects. Our analysis illustrates how the proximity of the stream's orbit to the 2:1 commensurability with Jupiter dominates the short-term variations in orbital elements.
INTRODUCTION
The Quadrantid meteor shower, first observed in 1835 (Fisher, 1930), occurs in the first week of January and although annual variations in the flux can be considerable (Mason, 1981) the shower is acknowledged to be one of the richest of the year (Hindley, 1970). The short duration of the shower ( - 2 days) probably implies that the stream is relatively young. Most observations of meteors come from three sources: visual, photographic, and radio. Measurements from these sources of the hourly rate o f meteors as a function of time result in an estimate of the time at which peak activity occurs, corresponding to the Earth's passage through the densest part of the stream. In the case of the Quadrantids, converting this time to a solar longitude is equivalent to measuring the longitude of ascending node of the stream's orbit. However, accurate values of individual meteor orbital elements can only be calculated using multiple-station photographic or radar measurements. All available visual observations of the longitude o f ascending node at the time of maximum activity are shown in Fig. 1. Using this data we estimate a linear regression
rate of - ~ 0 0 3 8 -+ ~0014 year -1 for the visual Quadrantids. In this paper we will show that the somewhat anomalous values for 1878 and 1979 may reflect real changes in the longitude of ascending node of the stream. Radio observations have been carried out since 1950 and Hughes (1972) estimated a regression rate of - ~ 0 0 4 1 _+ 02.0017 year -1 for the radio Quadrantids. A number of numerical studies have also been carried out involving the back integration of actual or mean Quadrantid orbits over several thousand years. The various estimates of the Quadrantid regression rate are summarized in Table I. In this paper we develop an analytic theory to account for the behavior shown in Fig. 1. We discuss the possibility that the smaller mass radio Quadrantids have a significantly different regression rate and show how near commensurabilities with Jupiter's period can affect observational and numerical estimates of the Quadrantid regression rate. NODAL REGRESSION
Consider the motion of two objects of masses m and m' with orbital elements a, e, I, fl, to, and a', e', I', i f , to', respectively, 125 0019-1035/82/010125-10502.00/0 CopyrightO 1982by AcademicPress, Inc. All rightsof reproductionin any formreserved.
126
C. D. MURRAY I
I
'
I
'
unchanged. The equation for the variation of the longitude of ascending node, II, is [see, for example, Brouwer and Clemence (1961, p. 284)]
I
283?4
-
dll dt
28.5?2
1 OU e2) u2 sin I 0 I '
(1)
where n is the mean motion o f m and U is the disturbing function. In a Cartesian frame with one disturbing body U is defined by
283~,C
.J
na2(l -
U = G m ' (1/A - r . r'/r'a),
282°.8
282"6
1 t850
,
,
i
I t890
,
, Yeor
,
1 4950
,
,
A I
,
1970
FIG. 1. Visual o b s e r v a t i o n s of the longitude of ascending node o f the Quadrantid meteor stream from 1843 to 1979. The data is taken from Hindley (1970, 1971), Millman (1976), and Sterne Weltraum (1979) 18, 305-307. The straight line represents a best fit linear regression analysis o f the data a n d c o r r e s p o n d s to a nodal regression rate of -0?.0038 +_ 07.0014 year -1.
where a is the semimajor axis, e the eccentricity, I the inclination, I~ the longitude of ascending node, and to the argument of perihelion. We will assume a' > a, m' >> m, and that the orbital elements o f m' remain
(2)
where A is the mutual separation of the bodies and r, r' denote their heliocentric position vectors. Since Eq. (2) is written in terms o f Cartesian coordinates it is necessary to transform to standard orbital elements before making use of Eq. (1). The expansion of the disturbing function in terms of orbital elements can be a tedious process. Peirce (1849) carried out a sixthorder development while Le Verrier (1855) managed to complete a seventh-order expansion. For most purposes the expansion of Brouwer and Clemence (1961, p. 490), which is complete to fourth order in eccentricity and inclination for the secular terms, is sufficient. If we consider purely secular terms in the expansion of Brouwer and Clemence and make the approximations e' = 0, I' = 0 then
TABLE I A SUMMARY OF PREVIOUS ESTIMATES OF THE QUADRANTID REGRESSION RATE IN COMPARISON WITH THE RESULTS OF THIS PAPER
Authors(s)
H a w k i n s and Southworth (1958) H a m i d and Y o u s s e f (1963) Hindley (1970) H u g h e s (1972) H u g h e s , Williams and Murray (1979) Murray (this paper)
Source/method
Perturbing planets
Observations (visual and radio) Numerical O b s e r v a t i o n s (visual) Observations (radio) Numerical O b s e r v a t i o n s (visual) Analytical (visual)
All Jupiter All All Jupiter, Earth All Jupiter, Saturn
(1 (°year -~) -0.006 - 0.0054 -0.0031 _+ 0.0004 -0.0041 _+ 0.0017 -0.00489 - 0 . 0 0 3 8 _+ 0.0014 -0.0041 _+ 0.0005
QUADRANTID NODAL REGRESSION the only relevant t e r m s in the e x p a n s i o n are
U = (Gm'/a') {-Ixr2otb~)2 + 8aoaa2t2b
btj )= 2s(s+ 1)(s + 2 ) . . . ( s + j -
o v e r c o m e the p r o b l e m of c o n v e r g e n c e can only be resolved b y reformulating the expansion as a Fourier series. Such an app r o a c h has b e e n used b y Kaula (1962). E m p l o y i n g for the m o s t part the notation o f Allan (1969), Kaula's e x p r e s s i o n is
(3)
where r~ = a/a', tr 2 = sin2½I, the b(,~) are Laplace coefficients defined b y
Gm' 2 a ~ a'
U-
l~2
l,l--2p
s(s + j) O/2 1)
I
x cos [(l - 2p)to + (l - 2p + q ) M - (l - 2p')oJ' - ( 1 - 2p'
1) o6
+ q')M' + m(l~ - fF)],
1.2.(j + 1)(j + 2)
(4)
IY' = dk / (dak) .
rm = 2 - 80,m, Fzmp(l) = it-"
Poole et al. (1972) give the appropriate m e a n Quadrantid elements as 3.07 A U , 0.681, 71°.4, 1702.4.
Differentiating Eq. (3) with respect to I and substituting in Eq. (I) with the a b o v e elements, we obtain l~ ~ +00.084 year -~ for the regression rate due to Jupiter's perturbations. This is both positive and greater than a n y of the o b s e r v e d values (see Table I) and clearly illustrates the inadequacy o f the standard m e t h o d w h e n applied to the Quadrantid stream. The failure of the method reflects either the need for a higherorder e x p a n s i o n or the difficulties in satisfying the c o n v e r g e n c e criteria o f the standard e x p a n s i o n [see, for example, Hagihara (1972)]. While the e x p a n s i o n s of Peirce and Le Verrier include terms o f higher order the algebra involved in their expressions bec o m e s unwieldy. Although this can be
(5)
where M and M' are the m e a n anomalies and
and D k is a differential o p e r a t o r defined b y
= = = =
~c-l-l,l-$10'(pt"
Xl_2p+q(e) "'l-2p,+q, -- , q,q r~--eo
1 +l.(j+
a e I to
F,,,~(I)Ft,,~,(I')
p~tv ~ 0
1)
+...],
(--1)l-mKm (1 -- m)! (l-+ m)!
!
X
+ s ( s + 1)(s + j ) ( s + j +
m=O
× ~,
1.2.3...j x ~
127
x
(6)
(1 + m)! 2tp!(l - p)[
l-m-
where k is f r o m m a x (0, l - m - 2p) to min (l - m, 21 - 2p) and c = cos ½I,
s = sin ½I.
(8)
The quantities Xg,b(e) are H a n s e n efficients (Plummer, 1918) defined by
co-
X~ "b (e) = (1 + B') -a-x
× 2
J=-®
J~ (ce)X~(e),
(9)
where # = e/[1 + (1 - e')ln],
(10)
the J~ (ce) are Bessel functions, and
X~,~(e) = ( - f l ) e-a-b ( a - b +d - 1) ×G(c-d-a - 1,-a -bc-d-b+ 1, fl2), d<-c-b,
1, (11)
128
C.D. MURRAY
X~J(e) = (_a)_,+d+ b ×G(-c
+d-a-
-c+d+b+
l, a2),
I -'> I*~
(_a+b+lc+d+b/~
1,-a +bd>-c-b,
r-->O,
where G(A, B, C, x) is the standard hypergeometric function defined by AB 1 + ~.lt.X
G(A,B,C,x)=
A ( A + 1)B(B + 1)x 2 + C(C + 1).2! + ....
fD""> ~, 0~ _..>~t,
1, (12)
(17)
where it is understood that ~ and o3' are now the longitudes of pericenter measured from the mutual node of the orbits (see Fig.
2). Since (13)
The distinct advantage of using Eq. (5) is that the expression is complete. Although this full expansion is intimidating, in practice it is only necessary to calculate the value of U or its derivatives for specific values of the integers l, m, p, q, p ' , and q'. These values are determined by the form of the argument ~b= ( 1 - 2p)ra+ ( 1 - 2p + q ) M (l - 2p')ra' - (l - 2p' + q')M' + m(fl - fl') (14)
Ftmp,(O) = 8t-2p,,raPlm(O),
(18)
where the P p are associated Legendre polynomials, we require 1 - 2p' = m = - q ' ,
(19)
In this reference frame we now have ~b = q ~ - q ' ~ ' .
(20)
-
of the relevant cosine term in Eq. (5).
SECULAR PERTURBATIONS
The nodal regression shown in Fig. 1 reflects the secular change in the longitude of ascending node over a period of 136 years. We can isolate the secular terms in Eq. (5) by excluding all those terms containing the mean anomalies since those terms are of short period and do not produce a net secular effect. Therefore [see Eq. (14)]
From Eqs. (15) and (16) it is obvious that q + q', l - q, and l - q' must all be even. A sample of the possible values of q, q', m, l, p, and p' are shown in Table II. With these restrictions we can now obtain expressions for Fz,,,, Fzmp,, X i ~ q , and XFJ~t{~,~': z~-m(l + m)! Ftmp(l*) = 2tp!(l _ p)!
mln(l-m,l--q) ( ) ~ ( _ l) ~ I - q k=max(O,-q-m) k ( l + q ) C*zl-q-ra-zks*ra.C-q+Zk, ×
× l-m-k
it-m(l + m)!
l-
2 p + q = 0,
(15)
1 - 2p' + q' = 0 .
(16)
To simplify some of the algebra and yet retain the general character of our solution we choose the reference plane to lie in the orbital plane of the perturber m'. This is equivalent to performing the mappings
Fzrap,(O) = 2'p' !(l - p')l'
Xt'2~(e)
= (1 + fl2)-z-iXtdToq(e),
x-l-l'l-2pt[a /-2pt+qt \c p~, =
(1
+ #12)lXo,/o-l'-q'(et).
(21)
(22) (23) (24)
Combining Eqs. (21)-(24), differentiating with respect to I*, and substituting in Eqs.
QUADRANTID NODAL REGRESSION
129
(5) and (1), we have
=
dFl
m'
n
~
tCm(l -- m ) ! (1 + rn)! 22~o!(1 - p ) ! p ' !(1 - p ' ) !
otl+ I
(1 - eS)lnsin~l * lffilmin
x
see mln(l--m,l--q) ~
kffimax(O,-q-m)
( ) (
( _ i) k I - q k
) 1 + q 1 -
m
( m + q + 2k
-
-
2ls2)c*21-q-m-2ks
× (1 + fl2)-1-1X~',-~q(e) (1
where M o is the mass o f the Sun. When calculating 03 and 03' it is first necessary to define the direction o f the mutual nodes. The transformation o f reference planes shown in Fig. 2 is equivalent to a rotation through an angle ft' about the axis
+
[3'2)lX~lo-l'-q'(e'
) cos(qt3 - q'&'),
While Eq. (25) can easily be adapted for calculation by a c o m p u t e r the high eccentricity o f the Quadrantid stream poses particular problems. Szeto and L a m b e c k (1981) demonstrated that great care must be taken in calculating the H a n s e n coefficients for large values o f e . The slow convergence o f the series in a means that the inclination
TABLE SELECTION
p, p~, A N D
OF THE
I USED
IN THE
SECULAR
q
q'
q + q'
m
0 0 I 2 0 1 2 3 4
0 2 I 0 4 3 2 I 0
0 2 2 2 4 4 4 4 4
0 2 1 0 4 3 2 I 0
II
POSSIBLE
VALUES
CALCULATION
OF q, q',
m,
OF PURELY
PERTURBATIONS
p
1,2,3, 1,2,3, 1,2,3, 0, 1, 2, 2, 3, 4, 1,2,3, 0,1,2, 0,1,2, 0,1,2,•
p'
. . . .. . . .
. . . . . .
1,2,3 .... 0,1,2, 1,2,3, 1,2,3, 0, 1 , 2 , 0,1,2, 0,1,2, 1,2,3, 2, 3, 4,
I
2,4,6 .... 2, 4, 6, . 3,5,7,.. 2, 4, 6, . 4, 6, 8, . 3, 5, 7, . 2, 4 , 6 , . 3, 5, 7, . 4,6,8,•
. . . . . . .
(25)
perpendicular to the ecliptic followed by a rotation through an angle I ' about the line o f nodes o f m'. The line o f mutual nodes then intersects the orbit o f mass m at true a n o m a l y f , given b y
[cos to sin (ll' - II) - sin to cos(IF - fl) cos Ii] . f " = tan-I Lsin to sin (fl' - fl) - cos to cos(fl' fl) cos
A
*m+q+2k
k
(26)
terms are calculated by summing a finite alternating series: while the sum o f the ser e s is o f order unity the individual terms can take large positive or negative values for large values of I. In some cases this requires the use o f double or quadruple precision arithmetic. The H a r v a r d Meteor Program (Jacchia and Whipple, 1956) provided orbital elements for 26 meteors identified as Quadrantids. O f these visual meteors we chose to examine the Quadrantid orbits calculated by Jacchia and Whipple (1961) and Hawkins and Southworth (1961). The remaining orbits, those of M c C r o s k e y and Posen (1961), were not included in our study since they had less accurate orbital elements. Instead o f performing a perturbation analysis on a mean Quadrantid orbit, the relatively large spread in orbital elements o f the 13 meteors (see Table III) suggested that it would be more relevant to calculate the perturbations on individual
130
C.D.
MURRAY
S
/
/w'--~6Y
FIG. 2. A s c h e m a t i c r e p r e s e n t a t i o n of the transforma t i on from the ecliptic reference plane to that of the orbit of the p erturbing body m ' . The reference direction c ha nge s from Y, the first point of Aries, to the line of mutual nodes of the orbits of m and m ' . The longitudes of pericenter, i and ~ ' , are then m e a s u r e d from the line o f mutual nodes and I* denote s the mutual inclination.
T A B L E III THE TRAIL NUMBER, ORBITAL ELEMENTS, CALCULATED REGRESSION RATE, AND EXPECTED RESONANT AMPLITUDE FOR 13 VISUAL QUADRANTID METEORS Trail number
a (AU)
e
I (o)
~lj (Oyear_l)
lls (Oyear-i)
l~ (Oyear-1)
(Ail)~s (o)
9945 9953 9955 9974 9983 9985 9997 10006 9907 9928 9930 9968 9992
3.046 2.906 3.119 2.999 3.002 3.074 3.274 3.170 2.88 3.05 2.89 2.72 2.96
0.682 0.664 0.685 0.674 0.675 0.683 0.701 0.691 0.659 0.679 0.662 0.640 0.670
68.6 72.7 72.6 72.5 72.4 70.8 73.4 72.5 72. i 70.0 71.7 71.0 73.1
- 0.0048 - 0.0036 -0.0038 - 0.0037 -0.0038 - 0.0043 -0.0030 -0.0038 - 0.0036 -0.0044 - 0.0038 - 0.0036 - 0.0035
- 0.0004 - 0.0003 -0.0003 - 0.0003 -0.0003 - 0.0003 -0.0003 -0.0003 - 0.0003 -0.0003 -0.0003 - 0.0003 - 0.0003
- 0.0052 - 0.0039 -0.0041 - 0.0040 -0.0041 - 0.0046 -0.0033 -0.0041 - 0.0039 -0.0047 - 0.0041 - 0.0039 - 0.0038
0.068 0.034 0.11 0.050 0.051 0.080 6.8 0.18 0.032 0.069 0.033 0.020 3.1
Note.
The orbital data are t a k e n from J a c c h i a and Whipple (1961) and H a w k i n s and S o u t h w o r t h (1961). (1 is the theoretical r e g r e s s i o n rate due to the effects of Jupiter (J) and Saturn (S). The m e a n re gre s s i on rate is -07.0041 -+ 0°.0005 y e a r -1. (AII)r~B is the e x p e c t e d amplitude of the short-term oscillations in gl c a u s e d by the d o m i n a n t resonance. The v alues 60.8 and 30.1 are i n s t a n t a n e o u s values w hi c h might c ha nge significantly if a second-order perturbation theory is employed. These meteors are close to e x a c t resonance.
QUADRANTID NODAL REGRESSION meteors. For each of our sample of 13 Quadrantids we calculated the secular regression rates due to effects of Jupiter and Saturn using the method outlined above. Perturbations by the Earth and other planets are negligible, contributing at most 10-e degrees year -a to the regression rate. For our purposes it was sufficient to consider only the q = q' = 0 term (see Table II) in Eq. (25) since the terms with q, q' > 0 were found to be - 1 0 0 times smaller. The resulting regression rates are summarized in Table III. F r o m our results we calculated a mean regression rate of 1] = -0°.0041 _ 02.0005 year -~. This compares favorably with both the observations and numerical estimates (see Table I). While agreement is good between the calculated secular regression rate and the observed rate averaged o v e r 136 years it is still necessary to try to account for the short-period fluctuations that are evident in Fig. 1. To do this we must estimate the effects o f resonant perturbations on our chosen sample o f Quadrantid meteors.
pansion o f the disturbing function. We will now examine how our results are modified by a near commensurability between a meteor's orbital period and that of Jupiter. A commensurability occurs when sen - "on' ~ 0,
ll-
2p + q = ~:,
(28)
2p' + q' = "o.
(29)
Choosing the same reference frame as before, the argument ~b becomes ~b= (se - q ) ~ +
RESONANT PERTURBATIONS
(m_~_)
(27)
where ~: and "o are integers and n and n' are the mean motions of the meteor and Jupiter, repectively. Those terms in the expansion of the disturbing function which contain expressions o f the form ~:M - "oM' in the argument ~b will produce small divisors after integration of the differential equation for the variation in f l [Eq. (1)]. Therefore, to determine the effects of an individual commensurability it is only necessary to isolate the relevant terms. Thus, referring to the full expansion in Eq. (5), we only require those terms where
-
So far we have only considered the effects o f the purely secular terms in the ex-
131
~:M ("o -
q') ~'
-
"oM',
(3O)
where q + q' - ~: - "o is always even. The resulting expression for d I l / d t is
n ~ - m)! ( 1 + m)! ( 1 - e2)V2sin2I * ctt+l 22;3((// - p ) ! p ' ! ( I - p')~ l=2
k = rnax ( O,--q + #--ra )
m
× (m + q - ~ + 2k - 21sZ)c*2t-q+e-m-2es *m+q-e+2k
x (1 + B2)-1-1 2
J~(~e) XejZ'e-q(e)
j=--r~
x (1 + B'2) 1 2
Jj,('oe') X -"O,j' " ' ~ - " ' ( e ')
× cos[(~¢ - q)& + ~ M -
('O - q')&' - "OM'].
(31)
132
C. D. MURRAY
If we write Eq. (31) as (--~)
= S cos ~b
(32)
and, as a first approximation, assume that S is independent of time, then S fires = f~0 + ~:n - ,/n' sin ~b = I~0 + (AI))~s sin ~b,
(33)
where II0 is a constant o f integration. Therefore (ZMl)~s is a measure o f the shortperiod change in fl to be expected due to the ~ : ¢ commensurability. While S itself may be small, if the meteor is close to an exact resonance (i.e., s r n - ~ n ' ~ 0) then large amplitude oscillations may result. The period o f such oscillations is given by Tr~s = 2~r/(¢n - 7qn') years,
(34)
where the mean motions are in units o f radians per year. A perfect example of this process in action is in the results of numerical integrations of a mean Quadrantid orbit carried out by Hughes et al. (1979). Their plot of 1~ against time showed a distinct periodicity combined with the linear trend in the variation o f the node. The period of the oscillations was - 6 0 years with an amplitude -00.13. By inserting the appropriate orbital elements for Jupiter and the mean Quadrantid orbit in Eq. (31) we calculate that the 2:1 commensurability with Jupiter would produce a 58-year periodicity with an amplitude o f 00.1 in the variation of f/. Hughes et al. (1979, 1980) incorrectly attributed the oscillations to the l l : 5 commensurability at 3.075 AU, close to the mean Quadrantid semimajor axis at 3.07 AU. This commensurability would have given rise to a 397year periodicity with an amplitude o f only 00.01. To estimate the effects o f resonant perturbations on the stream as a whole we calculated the amplitude o f the perturbations produced by e v e r y Jupiter resonance (up to and including order 7) on each of the
13 meteors in our sample. It was found that the 2:1 commensurability produced the dominant effect on all but one of the meteors regardless of the location of the meteor orbit. The associated values of (Aft)res are listed in Table III. Meteors No. 9997 and 9992 have anomalously large amplitude oscillations because of their proximity to the 2 : 1 and 7 : 3 resonances. By averaging (Al-/)~s for all other meteors we derive an estimate of the expected short-term variations in f / f o r the Quadrantid stream (Af~)res = 0°.066 _+ 00.046.
(35)
This compares favorably with the observations (see Fig. 1).
DISCUSSION By including the most recent observations we have calculated a new regression rate o f -0?.0038 -+ ~0014 year -1 for the Quadrantid meteor stream. Using a Fourier expansion of the disturbing function and considering the perturbations due to Jupiter and Saturn on individual meteors we have derived a theoretical regression rate of -07.0041 +_ 0?.0005 year -1 in good agreement with the observations. Furthermore, by investigating the effects o f resonant perturbations on the meteors we have shown that the 2 : 1 commensurability with Jupiter dominates the short-term behavior of the stream. This is borne out by both numerical experiments (Hughes et al., 1979) and visual observations. In particular the effects of resonance may explain the anomalous values o f the Quadrantid longitude o f ascending node recorded in 1878 and 1979. Radio observations of the Quadrantids have revealed that the smaller mass radio meteors intersect the Earth's orbit before the visual meteors. Analysis o f their orbital elements (Sekanina, 1970; Hughes et al., 1980) shows that the radio Quadrantids tend to have orbits with larger semimajor axes and eccentricities but that the spread
QUADRANTID NODAL REGRESSION in their elements is considerable. Possible explanations of this phenomenon have been put forward by Hughes et al. (1981). Although this paper has only been concerned with visual Quadrantids we do note that the larger values of a and e for the radio meteors imply a larger regression rate and that the spread in their elements could make the effects of the low-order Jupiter resonances beyond 3.3 AU appreciable. This could explain the consistently large fluctuations ( - i f 3 ) in observed values of II for the radio Quadrantids (Hughes, 1972). This work, the first analytic approach to the problem of nodal regression of the Quadrantid stream, illustrates the necessity of considering individual meteor orbits when investigating the behavior of a meteor stream. The use by many authors of a "mean orbit" is misleading when conclusions are drawn concerning alleged clustering of orbital elements in the past (Williams et al., 1979) or dispersion of orbits in the present (Murray et al., 1980). Although the mean semimajor axis of the Quadrantids in our sample is 3.07 AU, the orbits lie in the range 2.72 AU -< a - 3.274 AU. This, combined with similar spreads in e and I, results, in a variety of values for (1 and (Al'~)~s (see Table III). It is a combination of these effects which is reflected in the annual observations of the Quadrantid stream. While the Quadrantid meteors in our sample cover a broad range of semimajor axes it is interesting to note that all the meteors lie within the 2:1 resonance with Jupiter. We believe that the distribution of all meteors in the range 2 AU < a < 5 AU is significantly nonrandom and that Jovian resonances are largely responsible for determining local variations in meteor number density (Dermott and Murray, in preparation).
ACKNOWLEDGMENTS I wish to thank Stan Dermott and Frangois Mignard for helpful discussions. This work was supported by NSF Grant AST-8024042.
133
REFERENCES ALLAN, R. R. (1969). Evolution of the Mimas-Tethys commensurability. Astron. J. 74, 497-506. BROUWER, n . , AND G. M. CLEMENCE (1961). Methods o f Celestial Mechanics. Academic Press, New York. FISHER, W. J. (1930). The Quadrantid Meteors; history to 1927. Circ. Harvard College Obs., No. 346. HAGIHARA, V. (1972). Celestial Mechanics, Vol. II, Pt. I, pp. 58-79. MIT Press, Cambridge. HAMID, S. E., AND M. YOUSSEF (1963). A short note on the origin and age of the Quadrantids. Smithsonian Contrib. Astrophys. 7, 309-311. HAWKINS, G. S., AND R. B. SOUTHWORTH (1958). The regression of the node of the Quadrantids. Smithsonian Contrib. Astrophys. 3, 1-5. HAWKINS, G. S., AND R. B. SOUTHWORTH (1961). Orbital elements of meteors. Smithsonian Contrib. Astrophys. 4, 85-95. HINDLEY, K. B. (1970). The Quadrantid meteor stream in 1970. J. Brit. Astron. Assoc. 80, 479-486. HINDLEY, K. B. (1971). The Quadrantid meteor stream. J. Brit. Astron. Assoc. 82, 57-64. HUGHES, D. W. (1972). Nodal retrogression of the Quadrantid meteor stream. Observator), 92, 41-43. HUGHES, D. W., I. P. WILLIAMS, AND C. D. MURRAY (1979). The orbital evolution of the Quadrantid meteor stream between AD 1830 and 2030. Mort. Not. Roy. Astron. Soc. 189, 493-500. HUGHES, D. W., I. P. WILLIAMS, AND C. D. MURRAY (1980). The Quadrantid meteor stream: Past, present and future. In Solid Particles in the Solar System (I. Halliday and B. Macintosh, Eds.), pp. 153-156. Reidel, Dordrecht. HUGHES, D. W., I. P. WILLIAMS, AND K. Fox (1981). The mass segregation and nodal retrogression of the Quadrantid meteor stream. Mon. Not. Roy. Astron. Soc. 195, 625-637. JACCHIA, L. G., AND F. L. WHIPPLE (1956). The Harvard photographic meteor programme. Vistas Astron. 2, 982-994. JACCHIA, L. G., AND F. L. WHIPPLE (1961). Precision orbits of 413 photographic meteors. Smithsonian Contrib. Astrophys. 4, 97-129. KAULA, W. M. (1962). Development of the lunar and solar disturbing functions for a close satellite. Astron. J. 67, 300-303. LE VERRIER, U. J. J. (1855).Ann. Obs. Paris, Mdrn. 1, 71. MASON, J. (1981). The Quadrantid meteor shower. In 1981 Yearbook o f Astronomy. Norton, New York. McCROSKEY, R. E., AND A. POSEN (1961). Orbital elements of photographic meteors. Smithsonian Contrib. Astrophys. 4, 15-84. MILLMAN, P. M. (1976). Quadrantid meteors at 41,000 feet. Sky & Tel. 51, 225-228. MURRAY, C. D., D. W. HUGHES, AND I. P. WILLIAMS
134
C. D. M U R R A Y
(1980). The effect of orbital evolution on the influx of Quadrantid meteoroids. Mon. Not. Roy. Astron. Soc. 190, 733-741. PEmCE, B, (1849). Development of the perturbative function of planetary motion. Astron. J. 1, 1-8, 3136. PLUMMER, H. C. (1918). An lntroductoo, Treatise on Dynamical Astronomy, pp. 44-46. Cambridge Univ. Press, London. POOLE, L. M. G., D. W. HUGHES, AND T. R. KAISER (1972). Radio-echo observations of the major night-
time meteor streams, II1. Mon. Not. Roy. Astron. Soc. 156, 223-241. SEKANINA, Z. (1970). Statistical model of meteor streams. I. Analysis of the model. Icarus 13, 475494. SZETO, A., AND K. LAMBECK (1981). On eccentricity functions for eccentric orbits. Submitted for publication. WILLIAMS,I. P., C. D. MURRAY, AND D. W. HUGHES (1979). The long-term orbital evolution of the Quadrantid meteor stream. Mon. Not. Roy. Astron. Soc. 189, 483-492.
Nodal Regression of the Quadrantid Meteor Stream" An Analytic Approach C. D. MURRAY Center for Radiophysics and Space Research, Space Sciences Building, Cornell University, Ithaca, New York 14853 Received September 25, 1981; revised November 2, 1981 The mean orbit of the Quadrantid meteor stream has a high eccentricity and inclination with an aphelion close to the orbit of Jupiter. The nodal regression rate, a quantity which has been well determined from observations, cannot be calculated with sufficient accuracy using standard loworder expansions of the disturbing function. By using a high-order expansion of the disturbing function we show how the behavior of the longitude of ascending node of the Quadrantid stream is a result of both secular and resonant effects. Our analysis illustrates how the proximity of the stream's orbit to the 2:1 commensurability with Jupiter dominates the short-term variations in orbital elements.
INTRODUCTION
The Quadrantid meteor shower, first observed in 1835 (Fisher, 1930), occurs in the first week of January and although annual variations in the flux can be considerable (Mason, 1981) the shower is acknowledged to be one of the richest of the year (Hindley, 1970). The short duration of the shower ( - 2 days) probably implies that the stream is relatively young. Most observations of meteors come from three sources: visual, photographic, and radio. Measurements from these sources of the hourly rate o f meteors as a function of time result in an estimate of the time at which peak activity occurs, corresponding to the Earth's passage through the densest part of the stream. In the case of the Quadrantids, converting this time to a solar longitude is equivalent to measuring the longitude of ascending node of the stream's orbit. However, accurate values of individual meteor orbital elements can only be calculated using multiple-station photographic or radar measurements. All available visual observations of the longitude o f ascending node at the time of maximum activity are shown in Fig. 1. Using this data we estimate a linear regression
rate of - ~ 0 0 3 8 -+ ~0014 year -1 for the visual Quadrantids. In this paper we will show that the somewhat anomalous values for 1878 and 1979 may reflect real changes in the longitude of ascending node of the stream. Radio observations have been carried out since 1950 and Hughes (1972) estimated a regression rate of - ~ 0 0 4 1 _+ 02.0017 year -1 for the radio Quadrantids. A number of numerical studies have also been carried out involving the back integration of actual or mean Quadrantid orbits over several thousand years. The various estimates of the Quadrantid regression rate are summarized in Table I. In this paper we develop an analytic theory to account for the behavior shown in Fig. 1. We discuss the possibility that the smaller mass radio Quadrantids have a significantly different regression rate and show how near commensurabilities with Jupiter's period can affect observational and numerical estimates of the Quadrantid regression rate. NODAL REGRESSION
Consider the motion of two objects of masses m and m' with orbital elements a, e, I, fl, to, and a', e', I', i f , to', respectively, 125 0019-1035/82/010125-10502.00/0 CopyrightO 1982by AcademicPress, Inc. All rightsof reproductionin any formreserved.
126
C. D. MURRAY I
I
'
I
'
unchanged. The equation for the variation of the longitude of ascending node, II, is [see, for example, Brouwer and Clemence (1961, p. 284)]
I
283?4
-
dll dt
28.5?2
1 OU e2) u2 sin I 0 I '
(1)
where n is the mean motion o f m and U is the disturbing function. In a Cartesian frame with one disturbing body U is defined by
283~,C
.J
na2(l -
U = G m ' (1/A - r . r'/r'a),
282°.8
282"6
1 t850
,
,
i
I t890
,
, Yeor
,
1 4950
,
,
A I
,
1970
FIG. 1. Visual o b s e r v a t i o n s of the longitude of ascending node o f the Quadrantid meteor stream from 1843 to 1979. The data is taken from Hindley (1970, 1971), Millman (1976), and Sterne Weltraum (1979) 18, 305-307. The straight line represents a best fit linear regression analysis o f the data a n d c o r r e s p o n d s to a nodal regression rate of -0?.0038 +_ 07.0014 year -1.
where a is the semimajor axis, e the eccentricity, I the inclination, I~ the longitude of ascending node, and to the argument of perihelion. We will assume a' > a, m' >> m, and that the orbital elements o f m' remain
(2)
where A is the mutual separation of the bodies and r, r' denote their heliocentric position vectors. Since Eq. (2) is written in terms o f Cartesian coordinates it is necessary to transform to standard orbital elements before making use of Eq. (1). The expansion of the disturbing function in terms of orbital elements can be a tedious process. Peirce (1849) carried out a sixthorder development while Le Verrier (1855) managed to complete a seventh-order expansion. For most purposes the expansion of Brouwer and Clemence (1961, p. 490), which is complete to fourth order in eccentricity and inclination for the secular terms, is sufficient. If we consider purely secular terms in the expansion of Brouwer and Clemence and make the approximations e' = 0, I' = 0 then
TABLE I A SUMMARY OF PREVIOUS ESTIMATES OF THE QUADRANTID REGRESSION RATE IN COMPARISON WITH THE RESULTS OF THIS PAPER
Authors(s)
H a w k i n s and Southworth (1958) H a m i d and Y o u s s e f (1963) Hindley (1970) H u g h e s (1972) H u g h e s , Williams and Murray (1979) Murray (this paper)
Source/method
Perturbing planets
Observations (visual and radio) Numerical O b s e r v a t i o n s (visual) Observations (radio) Numerical O b s e r v a t i o n s (visual) Analytical (visual)
All Jupiter All All Jupiter, Earth All Jupiter, Saturn
(1 (°year -~) -0.006 - 0.0054 -0.0031 _+ 0.0004 -0.0041 _+ 0.0017 -0.00489 - 0 . 0 0 3 8 _+ 0.0014 -0.0041 _+ 0.0005
QUADRANTID NODAL REGRESSION the only relevant t e r m s in the e x p a n s i o n are
U = (Gm'/a') {-Ixr2otb~)2 + 8aoaa2t2b
btj )= 2s(s+ 1)(s + 2 ) . . . ( s + j -
o v e r c o m e the p r o b l e m of c o n v e r g e n c e can only be resolved b y reformulating the expansion as a Fourier series. Such an app r o a c h has b e e n used b y Kaula (1962). E m p l o y i n g for the m o s t part the notation o f Allan (1969), Kaula's e x p r e s s i o n is
(3)
where r~ = a/a', tr 2 = sin2½I, the b(,~) are Laplace coefficients defined b y
Gm' 2 a ~ a'
U-
l~2
l,l--2p
s(s + j) O/2 1)
I
x cos [(l - 2p)to + (l - 2p + q ) M - (l - 2p')oJ' - ( 1 - 2p'
1) o6
+ q')M' + m(l~ - fF)],
1.2.(j + 1)(j + 2)
(4)
IY' = dk / (dak) .
rm = 2 - 80,m, Fzmp(l) = it-"
Poole et al. (1972) give the appropriate m e a n Quadrantid elements as 3.07 A U , 0.681, 71°.4, 1702.4.
Differentiating Eq. (3) with respect to I and substituting in Eq. (I) with the a b o v e elements, we obtain l~ ~ +00.084 year -~ for the regression rate due to Jupiter's perturbations. This is both positive and greater than a n y of the o b s e r v e d values (see Table I) and clearly illustrates the inadequacy o f the standard m e t h o d w h e n applied to the Quadrantid stream. The failure of the method reflects either the need for a higherorder e x p a n s i o n or the difficulties in satisfying the c o n v e r g e n c e criteria o f the standard e x p a n s i o n [see, for example, Hagihara (1972)]. While the e x p a n s i o n s of Peirce and Le Verrier include terms o f higher order the algebra involved in their expressions bec o m e s unwieldy. Although this can be
(5)
where M and M' are the m e a n anomalies and
and D k is a differential o p e r a t o r defined b y
= = = =
~c-l-l,l-$10'(pt"
Xl_2p+q(e) "'l-2p,+q, -- , q,q r~--eo
1 +l.(j+
a e I to
F,,,~(I)Ft,,~,(I')
p~tv ~ 0
1)
+...],
(--1)l-mKm (1 -- m)! (l-+ m)!
!
X
+ s ( s + 1)(s + j ) ( s + j +
m=O
× ~,
1.2.3...j x ~
127
x
(6)
(1 + m)! 2tp!(l - p)[
l-m-
where k is f r o m m a x (0, l - m - 2p) to min (l - m, 21 - 2p) and c = cos ½I,
s = sin ½I.
(8)
The quantities Xg,b(e) are H a n s e n efficients (Plummer, 1918) defined by
co-
X~ "b (e) = (1 + B') -a-x
× 2
J=-®
J~ (ce)X~(e),
(9)
where # = e/[1 + (1 - e')ln],
(10)
the J~ (ce) are Bessel functions, and
X~,~(e) = ( - f l ) e-a-b ( a - b +d - 1) ×G(c-d-a - 1,-a -bc-d-b+ 1, fl2), d<-c-b,
1, (11)
128
C.D. MURRAY
X~J(e) = (_a)_,+d+ b ×G(-c
+d-a-
-c+d+b+
l, a2),
I -'> I*~
(_a+b+lc+d+b/~
1,-a +bd>-c-b,
r-->O,
where G(A, B, C, x) is the standard hypergeometric function defined by AB 1 + ~.lt.X
G(A,B,C,x)=
A ( A + 1)B(B + 1)x 2 + C(C + 1).2! + ....
fD""> ~, 0~ _..>~t,
1, (12)
(17)
where it is understood that ~ and o3' are now the longitudes of pericenter measured from the mutual node of the orbits (see Fig.
2). Since (13)
The distinct advantage of using Eq. (5) is that the expression is complete. Although this full expansion is intimidating, in practice it is only necessary to calculate the value of U or its derivatives for specific values of the integers l, m, p, q, p ' , and q'. These values are determined by the form of the argument ~b= ( 1 - 2p)ra+ ( 1 - 2p + q ) M (l - 2p')ra' - (l - 2p' + q')M' + m(fl - fl') (14)
Ftmp,(O) = 8t-2p,,raPlm(O),
(18)
where the P p are associated Legendre polynomials, we require 1 - 2p' = m = - q ' ,
(19)
In this reference frame we now have ~b = q ~ - q ' ~ ' .
(20)
-
of the relevant cosine term in Eq. (5).
SECULAR PERTURBATIONS
The nodal regression shown in Fig. 1 reflects the secular change in the longitude of ascending node over a period of 136 years. We can isolate the secular terms in Eq. (5) by excluding all those terms containing the mean anomalies since those terms are of short period and do not produce a net secular effect. Therefore [see Eq. (14)]
From Eqs. (15) and (16) it is obvious that q + q', l - q, and l - q' must all be even. A sample of the possible values of q, q', m, l, p, and p' are shown in Table II. With these restrictions we can now obtain expressions for Fz,,,, Fzmp,, X i ~ q , and XFJ~t{~,~': z~-m(l + m)! Ftmp(l*) = 2tp!(l _ p)!
mln(l-m,l--q) ( ) ~ ( _ l) ~ I - q k=max(O,-q-m) k ( l + q ) C*zl-q-ra-zks*ra.C-q+Zk, ×
× l-m-k
it-m(l + m)!
l-
2 p + q = 0,
(15)
1 - 2p' + q' = 0 .
(16)
To simplify some of the algebra and yet retain the general character of our solution we choose the reference plane to lie in the orbital plane of the perturber m'. This is equivalent to performing the mappings
Fzrap,(O) = 2'p' !(l - p')l'
Xt'2~(e)
= (1 + fl2)-z-iXtdToq(e),
x-l-l'l-2pt[a /-2pt+qt \c p~, =
(1
+ #12)lXo,/o-l'-q'(et).
(21)
(22) (23) (24)
Combining Eqs. (21)-(24), differentiating with respect to I*, and substituting in Eqs.
QUADRANTID NODAL REGRESSION
129
(5) and (1), we have
=
dFl
m'
n
~
tCm(l -- m ) ! (1 + rn)! 22~o!(1 - p ) ! p ' !(1 - p ' ) !
otl+ I
(1 - eS)lnsin~l * lffilmin
x
see mln(l--m,l--q) ~
kffimax(O,-q-m)
( ) (
( _ i) k I - q k
) 1 + q 1 -
m
( m + q + 2k
-
-
2ls2)c*21-q-m-2ks
× (1 + fl2)-1-1X~',-~q(e) (1
where M o is the mass o f the Sun. When calculating 03 and 03' it is first necessary to define the direction o f the mutual nodes. The transformation o f reference planes shown in Fig. 2 is equivalent to a rotation through an angle ft' about the axis
+
[3'2)lX~lo-l'-q'(e'
) cos(qt3 - q'&'),
While Eq. (25) can easily be adapted for calculation by a c o m p u t e r the high eccentricity o f the Quadrantid stream poses particular problems. Szeto and L a m b e c k (1981) demonstrated that great care must be taken in calculating the H a n s e n coefficients for large values o f e . The slow convergence o f the series in a means that the inclination
TABLE SELECTION
p, p~, A N D
OF THE
I USED
IN THE
SECULAR
q
q'
q + q'
m
0 0 I 2 0 1 2 3 4
0 2 I 0 4 3 2 I 0
0 2 2 2 4 4 4 4 4
0 2 1 0 4 3 2 I 0
II
POSSIBLE
VALUES
CALCULATION
OF q, q',
m,
OF PURELY
PERTURBATIONS
p
1,2,3, 1,2,3, 1,2,3, 0, 1, 2, 2, 3, 4, 1,2,3, 0,1,2, 0,1,2, 0,1,2,•
p'
. . . .. . . .
. . . . . .
1,2,3 .... 0,1,2, 1,2,3, 1,2,3, 0, 1 , 2 , 0,1,2, 0,1,2, 1,2,3, 2, 3, 4,
I
2,4,6 .... 2, 4, 6, . 3,5,7,.. 2, 4, 6, . 4, 6, 8, . 3, 5, 7, . 2, 4 , 6 , . 3, 5, 7, . 4,6,8,•
. . . . . . .
(25)
perpendicular to the ecliptic followed by a rotation through an angle I ' about the line o f nodes o f m'. The line o f mutual nodes then intersects the orbit o f mass m at true a n o m a l y f , given b y
[cos to sin (ll' - II) - sin to cos(IF - fl) cos Ii] . f " = tan-I Lsin to sin (fl' - fl) - cos to cos(fl' fl) cos
A
*m+q+2k
k
(26)
terms are calculated by summing a finite alternating series: while the sum o f the ser e s is o f order unity the individual terms can take large positive or negative values for large values of I. In some cases this requires the use o f double or quadruple precision arithmetic. The H a r v a r d Meteor Program (Jacchia and Whipple, 1956) provided orbital elements for 26 meteors identified as Quadrantids. O f these visual meteors we chose to examine the Quadrantid orbits calculated by Jacchia and Whipple (1961) and Hawkins and Southworth (1961). The remaining orbits, those of M c C r o s k e y and Posen (1961), were not included in our study since they had less accurate orbital elements. Instead o f performing a perturbation analysis on a mean Quadrantid orbit, the relatively large spread in orbital elements o f the 13 meteors (see Table III) suggested that it would be more relevant to calculate the perturbations on individual
130
C.D.
MURRAY
S
/
/w'--~6Y
FIG. 2. A s c h e m a t i c r e p r e s e n t a t i o n of the transforma t i on from the ecliptic reference plane to that of the orbit of the p erturbing body m ' . The reference direction c ha nge s from Y, the first point of Aries, to the line of mutual nodes of the orbits of m and m ' . The longitudes of pericenter, i and ~ ' , are then m e a s u r e d from the line o f mutual nodes and I* denote s the mutual inclination.
T A B L E III THE TRAIL NUMBER, ORBITAL ELEMENTS, CALCULATED REGRESSION RATE, AND EXPECTED RESONANT AMPLITUDE FOR 13 VISUAL QUADRANTID METEORS Trail number
a (AU)
e
I (o)
~lj (Oyear_l)
lls (Oyear-i)
l~ (Oyear-1)
(Ail)~s (o)
9945 9953 9955 9974 9983 9985 9997 10006 9907 9928 9930 9968 9992
3.046 2.906 3.119 2.999 3.002 3.074 3.274 3.170 2.88 3.05 2.89 2.72 2.96
0.682 0.664 0.685 0.674 0.675 0.683 0.701 0.691 0.659 0.679 0.662 0.640 0.670
68.6 72.7 72.6 72.5 72.4 70.8 73.4 72.5 72. i 70.0 71.7 71.0 73.1
- 0.0048 - 0.0036 -0.0038 - 0.0037 -0.0038 - 0.0043 -0.0030 -0.0038 - 0.0036 -0.0044 - 0.0038 - 0.0036 - 0.0035
- 0.0004 - 0.0003 -0.0003 - 0.0003 -0.0003 - 0.0003 -0.0003 -0.0003 - 0.0003 -0.0003 -0.0003 - 0.0003 - 0.0003
- 0.0052 - 0.0039 -0.0041 - 0.0040 -0.0041 - 0.0046 -0.0033 -0.0041 - 0.0039 -0.0047 - 0.0041 - 0.0039 - 0.0038
0.068 0.034 0.11 0.050 0.051 0.080 6.8 0.18 0.032 0.069 0.033 0.020 3.1
Note.
The orbital data are t a k e n from J a c c h i a and Whipple (1961) and H a w k i n s and S o u t h w o r t h (1961). (1 is the theoretical r e g r e s s i o n rate due to the effects of Jupiter (J) and Saturn (S). The m e a n re gre s s i on rate is -07.0041 -+ 0°.0005 y e a r -1. (AII)r~B is the e x p e c t e d amplitude of the short-term oscillations in gl c a u s e d by the d o m i n a n t resonance. The v alues 60.8 and 30.1 are i n s t a n t a n e o u s values w hi c h might c ha nge significantly if a second-order perturbation theory is employed. These meteors are close to e x a c t resonance.
QUADRANTID NODAL REGRESSION meteors. For each of our sample of 13 Quadrantids we calculated the secular regression rates due to effects of Jupiter and Saturn using the method outlined above. Perturbations by the Earth and other planets are negligible, contributing at most 10-e degrees year -a to the regression rate. For our purposes it was sufficient to consider only the q = q' = 0 term (see Table II) in Eq. (25) since the terms with q, q' > 0 were found to be - 1 0 0 times smaller. The resulting regression rates are summarized in Table III. F r o m our results we calculated a mean regression rate of 1] = -0°.0041 _ 02.0005 year -~. This compares favorably with both the observations and numerical estimates (see Table I). While agreement is good between the calculated secular regression rate and the observed rate averaged o v e r 136 years it is still necessary to try to account for the short-period fluctuations that are evident in Fig. 1. To do this we must estimate the effects o f resonant perturbations on our chosen sample o f Quadrantid meteors.
pansion o f the disturbing function. We will now examine how our results are modified by a near commensurability between a meteor's orbital period and that of Jupiter. A commensurability occurs when sen - "on' ~ 0,
ll-
2p + q = ~:,
(28)
2p' + q' = "o.
(29)
Choosing the same reference frame as before, the argument ~b becomes ~b= (se - q ) ~ +
RESONANT PERTURBATIONS
(m_~_)
(27)
where ~: and "o are integers and n and n' are the mean motions of the meteor and Jupiter, repectively. Those terms in the expansion of the disturbing function which contain expressions o f the form ~:M - "oM' in the argument ~b will produce small divisors after integration of the differential equation for the variation in f l [Eq. (1)]. Therefore, to determine the effects of an individual commensurability it is only necessary to isolate the relevant terms. Thus, referring to the full expansion in Eq. (5), we only require those terms where
-
So far we have only considered the effects o f the purely secular terms in the ex-
131
~:M ("o -
q') ~'
-
"oM',
(3O)
where q + q' - ~: - "o is always even. The resulting expression for d I l / d t is
n ~ - m)! ( 1 + m)! ( 1 - e2)V2sin2I * ctt+l 22;3((// - p ) ! p ' ! ( I - p')~ l=2
k = rnax ( O,--q + #--ra )
m
× (m + q - ~ + 2k - 21sZ)c*2t-q+e-m-2es *m+q-e+2k
x (1 + B2)-1-1 2
J~(~e) XejZ'e-q(e)
j=--r~
x (1 + B'2) 1 2
Jj,('oe') X -"O,j' " ' ~ - " ' ( e ')
× cos[(~¢ - q)& + ~ M -
('O - q')&' - "OM'].
(31)
132
C. D. MURRAY
If we write Eq. (31) as (--~)
= S cos ~b
(32)
and, as a first approximation, assume that S is independent of time, then S fires = f~0 + ~:n - ,/n' sin ~b = I~0 + (AI))~s sin ~b,
(33)
where II0 is a constant o f integration. Therefore (ZMl)~s is a measure o f the shortperiod change in fl to be expected due to the ~ : ¢ commensurability. While S itself may be small, if the meteor is close to an exact resonance (i.e., s r n - ~ n ' ~ 0) then large amplitude oscillations may result. The period o f such oscillations is given by Tr~s = 2~r/(¢n - 7qn') years,
(34)
where the mean motions are in units o f radians per year. A perfect example of this process in action is in the results of numerical integrations of a mean Quadrantid orbit carried out by Hughes et al. (1979). Their plot of 1~ against time showed a distinct periodicity combined with the linear trend in the variation o f the node. The period of the oscillations was - 6 0 years with an amplitude -00.13. By inserting the appropriate orbital elements for Jupiter and the mean Quadrantid orbit in Eq. (31) we calculate that the 2:1 commensurability with Jupiter would produce a 58-year periodicity with an amplitude o f 00.1 in the variation of f/. Hughes et al. (1979, 1980) incorrectly attributed the oscillations to the l l : 5 commensurability at 3.075 AU, close to the mean Quadrantid semimajor axis at 3.07 AU. This commensurability would have given rise to a 397year periodicity with an amplitude o f only 00.01. To estimate the effects o f resonant perturbations on the stream as a whole we calculated the amplitude o f the perturbations produced by e v e r y Jupiter resonance (up to and including order 7) on each of the
13 meteors in our sample. It was found that the 2:1 commensurability produced the dominant effect on all but one of the meteors regardless of the location of the meteor orbit. The associated values of (Aft)res are listed in Table III. Meteors No. 9997 and 9992 have anomalously large amplitude oscillations because of their proximity to the 2 : 1 and 7 : 3 resonances. By averaging (Al-/)~s for all other meteors we derive an estimate of the expected short-term variations in f / f o r the Quadrantid stream (Af~)res = 0°.066 _+ 00.046.
(35)
This compares favorably with the observations (see Fig. 1).
DISCUSSION By including the most recent observations we have calculated a new regression rate o f -0?.0038 -+ ~0014 year -1 for the Quadrantid meteor stream. Using a Fourier expansion of the disturbing function and considering the perturbations due to Jupiter and Saturn on individual meteors we have derived a theoretical regression rate of -07.0041 +_ 0?.0005 year -1 in good agreement with the observations. Furthermore, by investigating the effects o f resonant perturbations on the meteors we have shown that the 2 : 1 commensurability with Jupiter dominates the short-term behavior of the stream. This is borne out by both numerical experiments (Hughes et al., 1979) and visual observations. In particular the effects of resonance may explain the anomalous values o f the Quadrantid longitude o f ascending node recorded in 1878 and 1979. Radio observations of the Quadrantids have revealed that the smaller mass radio meteors intersect the Earth's orbit before the visual meteors. Analysis o f their orbital elements (Sekanina, 1970; Hughes et al., 1980) shows that the radio Quadrantids tend to have orbits with larger semimajor axes and eccentricities but that the spread
QUADRANTID NODAL REGRESSION in their elements is considerable. Possible explanations of this phenomenon have been put forward by Hughes et al. (1981). Although this paper has only been concerned with visual Quadrantids we do note that the larger values of a and e for the radio meteors imply a larger regression rate and that the spread in their elements could make the effects of the low-order Jupiter resonances beyond 3.3 AU appreciable. This could explain the consistently large fluctuations ( - i f 3 ) in observed values of II for the radio Quadrantids (Hughes, 1972). This work, the first analytic approach to the problem of nodal regression of the Quadrantid stream, illustrates the necessity of considering individual meteor orbits when investigating the behavior of a meteor stream. The use by many authors of a "mean orbit" is misleading when conclusions are drawn concerning alleged clustering of orbital elements in the past (Williams et al., 1979) or dispersion of orbits in the present (Murray et al., 1980). Although the mean semimajor axis of the Quadrantids in our sample is 3.07 AU, the orbits lie in the range 2.72 AU -< a - 3.274 AU. This, combined with similar spreads in e and I, results, in a variety of values for (1 and (Al'~)~s (see Table III). It is a combination of these effects which is reflected in the annual observations of the Quadrantid stream. While the Quadrantid meteors in our sample cover a broad range of semimajor axes it is interesting to note that all the meteors lie within the 2:1 resonance with Jupiter. We believe that the distribution of all meteors in the range 2 AU < a < 5 AU is significantly nonrandom and that Jovian resonances are largely responsible for determining local variations in meteor number density (Dermott and Murray, in preparation).
ACKNOWLEDGMENTS I wish to thank Stan Dermott and Frangois Mignard for helpful discussions. This work was supported by NSF Grant AST-8024042.
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REFERENCES ALLAN, R. R. (1969). Evolution of the Mimas-Tethys commensurability. Astron. J. 74, 497-506. BROUWER, n . , AND G. M. CLEMENCE (1961). Methods o f Celestial Mechanics. Academic Press, New York. FISHER, W. J. (1930). The Quadrantid Meteors; history to 1927. Circ. Harvard College Obs., No. 346. HAGIHARA, V. (1972). Celestial Mechanics, Vol. II, Pt. I, pp. 58-79. MIT Press, Cambridge. HAMID, S. E., AND M. YOUSSEF (1963). A short note on the origin and age of the Quadrantids. Smithsonian Contrib. Astrophys. 7, 309-311. HAWKINS, G. S., AND R. B. SOUTHWORTH (1958). The regression of the node of the Quadrantids. Smithsonian Contrib. Astrophys. 3, 1-5. HAWKINS, G. S., AND R. B. SOUTHWORTH (1961). Orbital elements of meteors. Smithsonian Contrib. Astrophys. 4, 85-95. HINDLEY, K. B. (1970). The Quadrantid meteor stream in 1970. J. Brit. Astron. Assoc. 80, 479-486. HINDLEY, K. B. (1971). The Quadrantid meteor stream. J. Brit. Astron. Assoc. 82, 57-64. HUGHES, D. W. (1972). Nodal retrogression of the Quadrantid meteor stream. Observator), 92, 41-43. HUGHES, D. W., I. P. WILLIAMS, AND C. D. MURRAY (1979). The orbital evolution of the Quadrantid meteor stream between AD 1830 and 2030. Mort. Not. Roy. Astron. Soc. 189, 493-500. HUGHES, D. W., I. P. WILLIAMS, AND C. D. MURRAY (1980). The Quadrantid meteor stream: Past, present and future. In Solid Particles in the Solar System (I. Halliday and B. Macintosh, Eds.), pp. 153-156. Reidel, Dordrecht. HUGHES, D. W., I. P. WILLIAMS, AND K. Fox (1981). The mass segregation and nodal retrogression of the Quadrantid meteor stream. Mon. Not. Roy. Astron. Soc. 195, 625-637. JACCHIA, L. G., AND F. L. WHIPPLE (1956). The Harvard photographic meteor programme. Vistas Astron. 2, 982-994. JACCHIA, L. G., AND F. L. WHIPPLE (1961). Precision orbits of 413 photographic meteors. Smithsonian Contrib. Astrophys. 4, 97-129. KAULA, W. M. (1962). Development of the lunar and solar disturbing functions for a close satellite. Astron. J. 67, 300-303. LE VERRIER, U. J. J. (1855).Ann. Obs. Paris, Mdrn. 1, 71. MASON, J. (1981). The Quadrantid meteor shower. In 1981 Yearbook o f Astronomy. Norton, New York. McCROSKEY, R. E., AND A. POSEN (1961). Orbital elements of photographic meteors. Smithsonian Contrib. Astrophys. 4, 15-84. MILLMAN, P. M. (1976). Quadrantid meteors at 41,000 feet. Sky & Tel. 51, 225-228. MURRAY, C. D., D. W. HUGHES, AND I. P. WILLIAMS
134
C. D. M U R R A Y
(1980). The effect of orbital evolution on the influx of Quadrantid meteoroids. Mon. Not. Roy. Astron. Soc. 190, 733-741. PEmCE, B, (1849). Development of the perturbative function of planetary motion. Astron. J. 1, 1-8, 3136. PLUMMER, H. C. (1918). An lntroductoo, Treatise on Dynamical Astronomy, pp. 44-46. Cambridge Univ. Press, London. POOLE, L. M. G., D. W. HUGHES, AND T. R. KAISER (1972). Radio-echo observations of the major night-
time meteor streams, II1. Mon. Not. Roy. Astron. Soc. 156, 223-241. SEKANINA, Z. (1970). Statistical model of meteor streams. I. Analysis of the model. Icarus 13, 475494. SZETO, A., AND K. LAMBECK (1981). On eccentricity functions for eccentric orbits. Submitted for publication. WILLIAMS,I. P., C. D. MURRAY, AND D. W. HUGHES (1979). The long-term orbital evolution of the Quadrantid meteor stream. Mon. Not. Roy. Astron. Soc. 189, 483-492.