Poynting-Robertson effect and capture of grains in exterior resonances with planets

0.13 MB·English

by  J. Klacka

Poynting-Robertson effect and capture of grains in
exterior resonances with planets
J. Klaˇcka and P. Pa´stor
Department of Astronomy,Physics of the Earth, and Meteorology,
Faculty of Mathematics, Physics and Informatics, Comenius University,
Mlynsk´a dolina, 842 48 Bratislava, Slovak Republic
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e-mail: [email protected]; [email protected]
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0
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n Abstract. Spherical meteoroids orbiting theSunexperienceorbital decay duetothePoynting-
a
J Robertsoneffectandcanbecometrappedincommensurabilityresonanceswithaplanet.Detail

9 conditionsfortrappingintheplanarcircularrestrictedthree-bodyproblemwithactionofsolar
1
electromagneticradiation onsphericalgrainarepresented.Theresultsaregivenintermsofthe

2
radiation presure factor β.
v
1 In general, the greater value of β, the shorter capture time in a resonance. Captures of
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6 particles may beevenseveral times longer than it is presented in literature. Analyticalformula
1
for maximum capture times is given.
1
4 Analytical theory for minimal capture eccentricities is presented. The obtained analytical
0
/ results differ from those presented in the literature. However, the known analytical results are
h
p not veryconsistent with ourdetail numerical computational results for β−values0.01 and 0.05
-
o and for the planet Earth. Minimum eccentricities for captures into resonances are also smaller,
r
t even in order of magnitude, than thepublished values.
s
a The presented results enable comparison with calculations for nonspherical particles.
:
v
i
X
Key words.meteoroids, electromagnetic radiation, celestial mechanics
r
a
1. Introduction
PhysicsofresonanceswithSolarSystemplanetsisdiscussedmainlyduringthelastthreedecades.The
orbital evolution of meteoroids near resonances with planets was also investigated. Besides gravita-
tionalforcestheeffectofsolarelectromagneticradiationinthe formofthe Poynting-Robertson(P-R)
effect is usually taken into account (Robertson 1937, Klaˇcka 2004, 2008a, 2008b), see e. g., Jackson
and Zook (1992), Weidenschilling and Jackson(1993),Marzariand Vanzani (1994), Sˇidlichovsky´ and
Nesvorny´ (1994), Liou and Zook (1995), Liou et al. (1995).
TheaimofourcontributionisdetailanalyticalreconsiderationoftheinfluenceoftheP-Reffectona
meteoroidwhich can become trapped in commensurability resonances with a planet. As a motivation
2 J. Klaˇcka and P. P´astor: Capture of grains in exterior resonances with planets
we can introduce that Weidenschilling and Jackson (1993) yields for theoretical minimal value of
eccentricityatthebeginningofcapturethevalue0.3506fordustparticlewithβ =0.01inresonance2:1
withtheEarth,whileoursimulationsshowthatrealvaluesmaybeevenlessthan0.1.Detailconditions
for trapping in the planar circular restrictedthree-body problem with action of solar electromagnetic
radiationonasphericalgrain(P-Reffectholds)arepresented.Weconcentrateonrealvaluesoforbital
elements – width of semimajor axis in resonance, minimal values of eccentricities at the beginning of
capture, and, capture time. We use for calculating osculating orbital elements simultaneously solar
gravityanddominantpartofsolarradiationpressure-term(centralaccelerationhaveform GM⊙(1
− −
β)e /r2). Results presented in this contribution may serve for comparisonof the results obtained for
R
sphericalandnonsphericaldustparticles(meteoroids),sinceP-Reffectholdsonlyifaspecialcondition
is fulfilled (see Eq. (9) in Klaˇcka 2008a; compare Eq. (73) in Klaˇcka 2008b) and real nonspherical
particles do not fulfill the condition.
2. Model – Equation of motion
All our numerical simulations are based on equation of motion written in the form (see Eq. (72) in
Klaˇcka 2008bfor more details)
dv = GM⊙ e Gm r − rP + rP
dt − r2 R− P r r 3 r 3
(cid:26)| − P| | P| (cid:27)
+β GM⊙ 1 v·eR e v ,
r2 − c R − c
n(cid:16) (cid:17) A′ m2o
e r/r ; β =7.682 10−4 Q′ , (1)
R ≡ | | × pr m [kg]
(cid:2) (cid:3)
where r is position vector of the particle (with respect to the Sun) and r is position vector of
P
the planet which moves in circular orbit around the Sun; mP is mass of the planet, M⊙ is mass of
the Sun, m is mass of the particle/meteoroid, A′ =π a′2, a′ is radius of the spherical particle, Q′ is
pr
dimensionlessefficiencyfactor(see,e.g.,vandeHulst1981).β =5.761 10−5Q′ /(̺[g/cm3]s[cm]),
× pr
forhomogeneoussphericalparticleintheSolarSystem,̺ismassdensityandsisradiusofthesphere.
Planar motion will be considered.
We use osculating orbital elements which considers solar gravity and dominant part of solar radi-
ationpressure-term,simultaneously:centralacceleration GM⊙(1 β)eR/r2 is usedin calculations
− −
of orbital elements and the corresponding quantities will be denoted with a subscript β.
J. Klaˇcka and P. P´astor: Capture of grains in exterior resonances with planets 3
3. Perturbation equations of celestial mechanics
Rewriting Eq. (1) into the form
dv = GM⊙(1−β) e Gm r − rP + rP
dt − r2 R− P r r 3 r 3
(cid:26)| − P| | P| (cid:27)
β GM⊙ v·eR e + v , (2)
− r2 c R c
(cid:16) (cid:17)
we can immediately write for perturbation acceleration to Keplerian motion
F = (F ) + (F ) ,
β β G β PR
r r r
(F ) = Gm − P + P ,
β G − P r r 3 r 3
(cid:26)| − P| | P| (cid:27)
(F ) = β GM⊙ v·eR e + v . (3)
β PR − r2 c R c
(cid:16) (cid:17)
Perturbationequationsofcelestialmechanicsyieldforosculatingorbitalelements(a –semimajor
β
axis;e – eccentricity;i – inclination (of the orbital plane to the reference frame); Ω – longitude of
β β β
the ascendingnode;ω –argumentofpericenter;Θ is the positionangleofthe particle onthe orbit,
β β
when measured from the ascending node in the direction of the particle’s motion, Θ =ω +f ):
β β β
da 2 a p
β β β
= F e sinf +F (1 + e cosf ) ,
dt 1 e2 µ(1 β) { β R β β β T β β }
− β r −
de p e +cosf
β β β β
= F sinf +F cosf + ,
β R β β T β
dt µ(1 β) 1+e cosf
r − (cid:26) (cid:20) β β(cid:21)(cid:27)
di r
β
= F cosΘ ,
β N β
dt µ(1 β)p
β
−
dΩ r sinΘ
β = p F β ,
β N
dt µ(1 β)pβ siniβ
−
dω 1 p 2+e cosf
β p β β β
= F cosf F sinf
β R β β T β
dt − e µ(1 β) − 1+e cosf
β r − (cid:26) β β (cid:27)
r sinΘ
β
F cosi ,
β N β
− µ(1 β)pβ siniβ
−
dΘβ µp(1 β)pβ r sinΘβ
= − F cosi , (4)
dt p r2 − µ(1 β)pβ β N siniβ β
−
p
where µ GM⊙ and r =pβ/(1+eβcosfβ); Fβ R, Fβ T and Fβ N are radial, transversaland normal
≡
components of perturbation acceleration.
4 J. Klaˇcka and P. P´astor: Capture of grains in exterior resonances with planets
3.1. P-R effect and perturbation equations of celestial mechanics
This subsection follows considerations presented in Klaˇcka (2004 – Sec. 6.1), or, in Klaˇcka (1992).
On the basis of Eq. (3), we can immediately write for components of perturbation acceleration to
Keplerian motion:
µ v µ v
β R β T
(F ) = 2 β , (F ) = β , (F ) =0 , (5)
β R PR − r2 c β T PR − r2 c β N PR
where µ GM⊙, r=pβ/(1+eβcosfβ) and two-body problem yields
≡
v = µ (1 β)/p e sinf ,
β R β β β
−
q
v = µ (1 β)/p (1+e cosf ) . (6)
β T β β β
−
q
Inserting Eqs. (5) – (6) into Eq. (4), one easily obtains
daβ µ 2aβ 1+e2β +2eβcosfβ +e2βsin2fβ
= β ,
dt − r2 c 1 e2
(cid:18) (cid:19)PR − β
de µ 1
β = β 2e +e sin2f +2cosf ,
dt − r2 c β β β β
(cid:18) (cid:19)PR
di (cid:0) (cid:1)
β
= 0 ,
dt
(cid:18) (cid:19)PR
dΩ
β
= 0 ,
dt
(cid:18) (cid:19)PR
dω µ 1 1
β
= β (2 e cosf )sinf ,
dt − r2 c e − β β β
(cid:18) (cid:19)PR β
dΘ µ(1 β)p
β β
= − . (7)
dt r2
(cid:18) (cid:19)PR p
3.2. Gravity of a planet and perturbation equations of celestial mechanics
As for gravitational acceleration generated by a planet (F ) presented in Eq. (3), we can write
β G
(compare Eqs. (44) in Klaˇcka 2004 for the case i = Ω = ω = 0 and Eqs. (57) in Klaˇcka 2004)
P P P
r = r e , r =r e ,
P P PR R
e = (cosΘ ,sinΘ ,0) ,
PR P P
e = (cosΘ ,sinΘ ,0) ,
R β β
e = ( sinΘ ,cosΘ ,0) ,
T β β
−
e = cos(Θ Θ ) e + sin(Θ Θ ) e ,
PR P β R P β T
− −
ΘP = nP(t − t0) + ΘP0 , nP = G (M⊙ + mP) rP−3/2 . (8)
p
J. Klaˇcka and P. P´astor: Capture of grains in exterior resonances with planets 5
Inserting Eqs. (8) into (F ) presented in Eq. (3), we obtain
β G
(F ) = Gm (X X ) ,
β G P I − II
[r cos(Θ Θ ) r] e + r sin(Θ Θ ) e
P P β R P P β T
X = − − − ,
I [r2 + r2 2 r r cos(Θ Θ )]3/2
P − P P − β
1
X = [cos(Θ Θ ) e + sin(Θ Θ ) e ] . (9)
II r2 P − β R P − β T
P
In order to obtain (da /dt) , (de /dt) , ... (dΘ /dt) , it is sufficient to put Eqs. (9) into Eqs. (4).
β G β G β G
4. Exterior mean motion resonances and da /dt
β
We will consider only terms proportional to e0 and e1, i. e., we will neglect e2 and higher orders in
β β β
e . First of Eqs. (4) can be written in the form
β
da a
β β
= 2 a F e sinf +F (1 + e cosf ) . (10)
β β R β β β T β β
dt µ(1 β) { }
r −
On the basis of Eqs. (9) we write, then (µ Gm ):
P P
≡
r cos(Θ Θ ) a cos(Θ Θ )
(F ) = µ P P − β − β P − β ,
β R G P X(0) − r2
(cid:26) P (cid:27)
r sin(Θ Θ ) sin(Θ Θ )
(F ) = µ P P − β P − β ,
β T G P X(e ) − r2
(cid:26) β P (cid:27)
X(e ) = r2 +a2 (1 2e cosf ) 2a r (1 e cosf )cos(Θ Θ ) 3/2 , (11)
β P β − β β − β P − β β P − β
(cid:2) (cid:3)
for radial and transversalcomponents of gravitationalperturbation acceleration.
Aparticleisinexteriormeanmotionresonancewithaplanetwhentheratiooftheirmeanmotions
is approximately the ratio of two small natural numbers: n /n =j/(j+q) corresponds to q order
β P
−
exterior resonance. For the first-order resonances we have n /n = j/(j +1), i. e., any first-order
β P
resonance may be defined by the ratio T/T =(j+1)/j.
P
As for commensurability resonances with a planet moving in the circular orbit, the third Kepler’s
law [a3βn2β =µ(1−β), a3Pn2P =µ(1+mP/M⊙)] yields
2/3 −1/3
a T m
β =(1 β)1/3 1 + P , (12)
aP − (cid:18)TP(cid:19) (cid:18) M⊙(cid:19)
forthesemimajoraxisandperiodofrevolutionaroundtheSunoftheparticleandtheplanet(subscript
P). Defining any resonance by the ratio T/T , Eq. (12) yields immediately the ratio a /a .
P β P
On the basis of Eqs. (11) we can write for the important resonant terms
RES (F ) e sinf + (F ) (1+e cosf ) ,
≡ β R G β β β T G β β
r a r
(cid:8) P β (cid:9)P
RES = µ e sin(Θ ω ) e sinf + sin(Θ Θ ) . (13)
P β P β β β P β
X(0) − − X(0) X(e ) −
(cid:26) β (cid:27)
6 J. Klaˇcka and P. P´astor: Capture of grains in exterior resonances with planets
Using the relation
1 ε cos(Θ Θ )
P β
X(e ) = X(0) 1 3 e cosf − − ,
β − β β 1 2 ε cos(Θ Θ )+ε2
(cid:26) − P − β (cid:27)
ε r /a , (14)
P β
≡
We can Eq. (13) rewrite to the following form
µ e
P β
RES = sin(Θ ω )
a2β {1−2εcos(ΘP −Θβ)+ε2}5/2 (− β− β
7 1
+ ε sin(Θ ω ) sin(2Θ Θ ω )
P β β P β
2 − − 2 − −
(cid:20) (cid:21)
7 3
+ ε2 sin(Θ 2Θ +ω )+ sin(3Θ 2Θ ω ) 2sin(Θ ω )
β P β β P β β β
4 − 4 − − − −
(cid:20) (cid:21)
+ ε3 sin(Θ ω ) . (15)
P β
− )
Inthisexpressionweneglecttermsproportionaltosin(Θ Θ ),becausetheyhavenotlargeinfluence
β P
−
when we take into account terms important in first-order mean motion resonance. We use Laplace
coefficients b defined by the relation
j
+∞
1
(1 2εcos(Θ Θ )+ε2)−5/2 = b cosj(Θ Θ ). (16)
β P j β P
− − 2 −
−∞
X
From Eq. (15) using Eq. (16) we get for the first-order exterior resonance
µ e 1 1 7 1 7 3
RES = Pa2 β 2 −2 bj +ε 4 bj+1− 4 bj−1 +ε2 −bj − 8 bj+2+ 8 bj−2
β (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)
ε3
+ bj+1 sin[(j+1)Θβ jΘP ωβ] , (17)
2 − −
(cid:21)
where we have taken into account only terms with sine argument (j +1)Θ jΘ ω = Γ. We
β P β
− −
define C as
j
1 1 7 1 7 3
Cj bj +ε bj+1 bj−1 +ε2 bj bj+2+ bj−2
≡ − j+1 −2 4 − 4 − − 8 8
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)
ε3
+ bj+1 . (18)
2
(cid:21)
Using Eqs. (10), (13), (17) and (18) we obtain
da µ
β P
= (j+1) a e n C sinΓ . (19)
β β β j
dt − µ(1 β)
(cid:18) (cid:19)G −
For the first-order exterior mean motion resonance Eq. (12) reduces to
a j + 1 2/3 m −1/3
β =(1 β)1/3 1 + P . (20)
aP − (cid:18) j (cid:19) (cid:18) M⊙(cid:19)
J. Klaˇcka and P. P´astor: Capture of grains in exterior resonances with planets 7
Table 1. Values of C for particle with β = 0.01 and β = 0.05 in first-order exterior mean motion
j
resonance with the Earth in a circular orbit.
j Cj(β =0.01) Cj(β =0.05)
1 1.31819 1.35328
2 1.68270 1.77804
3 2.05585 2.23845
4 2.43605 2.73560
5 2.82257 3.27183
6 3.21514 3.85027
7 3.61366 4.47467
8 4.01813 5.14933
9 4.42856 5.87920
10 4.84502 6.66992
11 5.26757 7.52797
12 5.69630 8.46078
13 6.13128 9.47693
14 6.57262 10.58637
The values of C are given in Tab. 1 for particles with β 0.01,0.05 in the first-order exterior
j
∈ { }
resonance j 1,2,3,...,14 with planet Earth in circular orbit.
∈{ }
Neglecting higher orders of eccentricity e , the first of Eqs. (7) yields
β
da µ
β
= 2 β (1 + 4e cosf ) . (21)
β β
dt − a c
(cid:18) (cid:19)PR β
Eqs. (3) and (4) give
da da da
β β β
= + . (22)
dt dt dt
(cid:18) (cid:19)G (cid:18) (cid:19)PR
Exterior resonances can produce long-livedtrapping: da /dt= 0. On the basis of Eqs. (19), (21) and
β
(22) we can write
µ µ
P
(j+1) a e n C sinΓ=2 β (1 + 4e cosf ) . (23)
β β β j β β
− µ(1 β) a c
β
−
Relation nβ = nP j/(j +1) and the last one of Eqs. (8) yield nβ = √µ (1 + mP/M⊙)1/2 aP−3/2
j/(j+1). Using this result and also Eq. (20) in Eq. (23),
−1
1µP c (1+mP/M⊙)−1/6(j+1)4/3
e = 4( cosf )+ C ( sinΓ) . (24)
β ( − β 2 µ µ/aP β(1 β)1/3 j1/3 j − )
−
p
8 J. Klaˇcka and P. P´astor: Capture of grains in exterior resonances with planets
Eq. (24) immediately yields criterion for resonant trapping:
−1
1 µP c (1+mP/M⊙)−1/6 (j+1)4/3
e = 4 + C . (25)
β min ( 2 µ µ/aP β(1 β)1/3 j1/3 j)
−
p
5. Comparison of analytical and numerical results
WehavenumericallysolvedEq.(1)forvariousvaluesofβ andforvariousinitialconditions.Themost
important results are presented in Tables 2 and 3 for the case of the first-order exterior resonances
with the Earth. Numerical results are compared with analytical results obtained from the theory
presented in Secs. 2-4 and, also,with the analytical results presented by Weidenschilling and Jackson
(1993). Our theoretical values of semimajor axis for the exterior resonance (j +1)/j are obtained
from Eq. (20). The values are given in columns labeled as a . The values a and a are
β β min β max
minimal and maximal semimajor axes during the capture in a given resonance. The width of the
resonances in semimajor axis can be calculated from the relation a a . Minimal values of
β max β min
−
eccentricities for the capture into resonances,found by numerical solution of Eq. (1), are given in the
column e . Minimal capture eccentricities e were computed from Eq.(25). Minimal
β min num. β min th.
capture eccentricities e , presented in Tab. 2, are taken from Table II in Weidenschilling
β min th. WJ
and Jackson (1993). Maximal capture times for different first-order resonances are also given. The
times and the minimum and maximum semimajor axis values are obtained for a given numerical
integration of a meteoroid. The extremal values of semimajor axis occur when the maximal capture
time exists, for a given resonance.The minimum eccentricity values were found independently. There
is no relation between the minimum eccentricity and the set maximal capture times, minimum and
{
maximum semimajor axis values , as for the presentations in Tables 2 and 3.
}
GeneraltrendseenfromTables2and3isthatthelargervalueofj –typeofthefirst-orderexterior
resonances with the Earth is (j +1)/j, the smaller width of resonances in semimajor axes and the
shorter capture times. Moreover, the greater value of β, i) the smaller resonant width in semimajor
axes, and, ii) the shorter capture time.
6. Extrapolation of capture times
We found, from numericalintegrationsof Eq.(1), that maximalcapture time for a particle in a given
meanmotionresonanceisproportionaltothevalueoftheparticle’sβ andsemimajoraxisa according
β
to
a2
τ β . (26)
∼ β
J. Klaˇcka and P. P´astor: Capture of grains in exterior resonances with planets 9
Table 2.Characteristicsforthefirst-order(j+1)/j resonanceswithEarthforparticlewithβ =0.01.
j a a a e e e maximal
β β min β max β min β min β min
num. th. th. WJ capturetime
[ AU ] [ AU] [ AU ] [ 103 years ]
1 1.5821 1.5761 1.5881 0.102 0.1107 0.3506 1492
2 1.306 1.3023 1.3097 0.036 0.0784 0.0387 736
3 1.2074 1.2046 1.2101 0.011 0.0564 0.0227 742
4 1.1565 1.1544 1.1586 0.008 0.0418 0.0148 429
5 1.1255 1.1238 1.1272 0 0.032 0.0104 268
6 1.1045 1.1031 1.1059 0.003 0.025 0.0077 252
7 1.0894 1.0881 1.0907 0.001 0.0201 0.0059 227
8 1.0781 1.077 1.079 0 0.0164 0.0046 362
9 1.0692 1.0681 1.0701 0.001 0.0136 0.0038 167
10 1.062 1.061 1.0626 0 0.0114 0.0031 469
11 1.0562 1.0551 1.0566 0.002 0.0097 0.0026 75
12 1.0513 1.0503 1.0516 0.001 0.0084 0.0022 36
13 1.0471 1.0459 1.0479 0.001 0.0073 0.0019 21
14 1.0436 1.0423 1.0444 0.001 0.0064 0.0016 13
This relation is also in accordance with the expected dependence of secular time derivation of eccen-
tricity(LiouandZook1997;Klaˇckaetal.2008–η 0)ona andβ.UsingEq.(12)(withassumption
P
≡
mP M⊙) we obtain from Eq. (26)
≪
τ(aP1,β1) aP1 2 β2(1 β1)2/3
= − . (27)
τ(aP2,β2) (cid:18)aP2(cid:19) β1(1−β2)2/3
Eq. (27) presents the analytical formula for obtaining maximum capture time τ(aP2,β2) for the
particle of β2 and planetary orbital radius aP2 for a given type of resonance, if maximum capture
time τ(aP1,β1) is known for for the particle of β1 and planetary orbital radius aP1 for the same type
of resonance.
Using Eq. (27) we have calculated the values of capture times in Tab. 4 from values of capture
times in Tables 2 and 3. We use the values in Tab. 2 in calculations of the expected values in Tab. 3,
and vice versa.
10 J. Klaˇcka and P. P´astor: Capture of grains in exterior resonances with planets
Table 3.Characteristicsforthefirst-order(j+1)/j resonanceswithEarthforparticlewithβ =0.05.
j a a a e e maximal
β β min β max β min β min
num. th. capturetime
[ AU] [ AU ] [ AU] [ 103 years ]
1 1.5605 1.5555 1.5656 0.196 0.1981 216
2 1.2882 1.2851 1.2911 0.096 0.1701 168
3 1.1909 1.1887 1.1931 0.061 0.1422 187
4 1.1407 1.1389 1.1426 0.045 0.1172 80
5 1.1101 1.1088 1.1114 0.028 0.096 56
6 1.0894 1.0879 1.091 0.018 0.0786 102
7 1.0746 1.0735 1.0756 0.016 0.0645 62
8 1.0633 1.0626 1.0643 0.01 0.0531 23
9 1.0546 1.0538 1.0552 0.007 0.044 20
10 1.0475 1.0466 1.0481 0.005 0.0366 13
11 1.0418 1.0408 1.0424 0.005 0.0307 6
12 1.0369 1.036 1.0376 0.004 0.0258 3
13 1.0328 1.0317 1.0334 0.005 0.0218 5
14 1.0293 1.028 1.0301 0 0.0185 1
7. Conclusion
The planar circularrestricted three-body problem with actionof solarelectromagnetic radiationona
spherical grain is considered. This Poynting-Robertsoneffect is studied numerically and analytically.
We have found minimal values of capture eccentricities into the first-order exterior mean motion
resonances. Unfortunately, our theory for the minimal values of the capture eccentricities cannot
explain the found numerically values. The numerical values e are also smaller than the values
β min
presented in Weidenschilling and Jackson (1993). Thus, any analytical method explaining minimal
values of capture eccentricities does not exist, for the first-order exterior mean motion resonances.
Ournumericalresultsofferseveralresonantcharacteristicsforβ values0.01and0.05forthefirst-
−
order exteriorresonanceswith the Earth;capture times for β = 0.1 are in orderof magnitude shorter
than for β = 0.05. We have succeeded in finding greater values of capture times than it is presented
in Sˇidlichovsky´ and Nesvorny´ (1994).Eq. (27)presents an analyticalformula for obtaining maximum
capture time for the particle of β2 and planetary orbital radius aP2 for a given type of resonance, if