Generalized functions and statistical problems of. orbital mechanics

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1 Genealized functions and statistical poblems of obital mechanics Meshcheyakov TSNIIMASH OSKOSMOS 4// 8th US/ussian Space Suveillance Wokshop,

2 Intoduction Thee is discussed a new method fo solution of statistical obital poblems which descibe the motion of an ensemble of paticles. The method is an altenative to the usually used Eule algoithms. The method can be educed to tansfomation of statistical distibutions, beginning fom distibutions descibed by genealized functions by Diac. Analytical fomulas ae obtained fo the steady flows of obital media and inteplanetay dust paticles in vicinity of the Eath, consideing the shadowing by the Eath. The numeical solution of the unsteady flow in vicinity of gavitating cente is studied. The gaph of the unsteady dag foce acting on a body at the dust egion is depicted. The esults can be used both in isk assessments fo SC and in some poblems of astophysics 4// 8th US/ussian Space Suveillance Wokshop,

3 Space density induced by a fixed paticle ρ ϕ, ω, δ sin Φ sin ϕ δ Ω ω δ Hee ϕ,ω, is a geocentic fame, ϕ is a latitude, ω is a longitude, is a adius-vecto of a pobe point, Φ, Ω, and is a latitude, a longitude, and a adius-vecto of a paticle. It is easy to check that thee is a eally distibution, i.e. it meets all the nomalization equiements. 4// 8th US/ussian Space Suveillance Wokshop,

4 Space density induced by a paticle aveaged ove a fixed obit The density aveaged ove the peiodic motion in a fixed obit: o ρ ϕ, ω,, ω,, t dt ρ ϕ T T ρ ϕ, ω, e πp δ sin Φ sin ϕ δ sin cosi i cos ω Ω whee T is an obital peiod, T π e p µ p is a focal paamete, 4// 8th US/ussian Space Suveillance Wokshop, 4

5 Space density induced by a paticle aveaged ove a fixed obit ΦΦω and ω ae analogs of the equations of motion Φ Φt and t sinφ sini sin ω Ω sin i cos ω Ω cosφ cosi sin i cos ω Ω p + ecos cos cosi cos ω Ω sin i cos ω Ω Ω is a longitude of ascending node, is an agument of peicente. 4// 8th US/ussian Space Suveillance Wokshop, 5

6 Space density induced by a paticle aveaged ove a obit with nodal pecession ρ ϕ, ω, ρ ϕ, ω,, T t dt o T e ρ ϕ, ω, δ + δ π p sin i sin T is a peiod of the nodal pecession, ϕ -i ϕ i, p + e cos p e cos cos sin i sin sini ϕ 4// 8th US/ussian Space Suveillance Wokshop, 6

7 Space density induced by a paticle aveaged ove a obit with a double nodal &apsidal pecession T ρ4 ϕ, ω, ρ ϕ, ω, tdt, T o ρ ϕ, ω, 4 π a sin i sin ϕ T is a a peiod of the pecession of a peicente, -i ϕ i, а + / is a majo half axis. It is a fomula by D. Kessle. 4// 8th US/ussian Space Suveillance Wokshop, 7

8 All above deived distibutions ae nomalized on a paticle, i.e. π π dω cos ϕdϕ d ρ ϕ, ω, π 4// 8th US/ussian Space Suveillance Wokshop, 8

9 Nondimensional paametes Speed, adius-vecto, impact paamete, space density, accelaation of the dust paticle, time, and a foce acting on the gavitating g cente: ~ v v ~ v ~ v χ χ v µ µ ~ v t t µ ~ ρ ~ M v ρ a a ρ mρ µ ~ v F F mρ µ Hee m is a mass of a dust paticle, M is a mass of the gavitating cente, G is a univesal gavitational constant, µ M G Initial intensity of the flow and incement of dust space density: f δ ~ ρ ρv 4// 8th US/ussian Space Suveillance Wokshop, 9

10 The flow of unidiectional meteooid steam in vicinity of the Eath Space density induced by a paticle distibuted unifomly ove azimuth ϕ azimuth ϕ,, δ δ ρ Θ t t t sin,, π ρ Θ t t 4// 8th US/ussian Space Suveillance Wokshop,

11 Aveaging ove a time inteval T g g o T dt t T,,, ρ ρ, δ ρ sin, i v T δ π ρ Thee is a summation is fulfiled ove the solutions of the equation of motion Thee ae two solutions and two coesponding impact paametes: Θ t + ± cos 4 sin sin, χ 4// 8th US/ussian Space Suveillance Wokshop,

12 Aveaged space density Angula velocity and flux of paticles v χ N πχ d χ T f Space density δ ρ, f dχ i sin o ρ, f i sin χ Hee d χ dχχ χ χ sin This solution numeically coesponds to the esult obtained ecently by J. Jones & L.M.G. Poole. 4// 8th US/ussian Space Suveillance Wokshop,

13 Gavitational focusing and shadowing by the Eath Thee ae given the flow of paticles with initial velocities of km/sec. The main featue: behind the Eath fo distances geate some citical distance thee is a singulaity ρ, f π The citical distance coesponds to the geostationay egion if the steam velocity is about 8. km/sec. 4// 8th US/ussian Space Suveillance Wokshop,

14 Density in the tansvesal plane f ρ Pofile of space density in the tansvesal plane Dependency between impact paamete and coss point χ c 4// 8th US/ussian Space Suveillance Wokshop, 4 Dependency between impact paamete and coss point + χ c

15 Isotopic flows Density of the flow without shadowing A citical distance f + Em Em ρ f + Pofiles of the dust density v5 km/sec. - without shadowing - consideing shadowing unidiectional steam in the tansvesal plane 4// 8th US/ussian Space Suveillance Wokshop, 5

16 Angula Dead zone Coss-section of absoption Usual coss-section of absoption : S π m Coss-section of absoption consideing the gavitation effect : S + S m 4// 8th US/ussian Space Suveillance Wokshop, 6

17 Gavitational dag in the infinite dusty space Initially t, dust paticles distibuted unifomly ove an infinite space. Gavitating cente momentaily gets a velocity taken as unity. Equations of motions of the dust paticles : t f,, g,, 4// 8th US/ussian Space Suveillance Wokshop, 7

18 Paticle motion in time in linea appoach without self gavitation effects Space density ρ,, t ρ,, t i i whee sin f ρ,, i t sin g f g f i i and ae the solutions of the above given equations of motion Contol scheme: 4// 8th US/ussian Space Suveillance Wokshop, 8

19 Time dependencies in linea appoach Incements of space density In two points at 9 Time dependency fo the dag foce 4// 8th US/ussian Space Suveillance Wokshop, 9

20 Conclusion The poposed method can be used in poblems of statistical celestial mechanics. Thee ae esolved seveal poblems on peiodic and apeiodic steady flows which ae impotant in SC isk analysis. Thee ae given analytical fomulas fo space density of meteooid dust in vicinity of the Eath consideing focusing and shadowing effects. The poblem of unsteady motion of gavitating body in a dust envionment is investigated and the gaph of unsteady dag in the dust media has been calculated. This wok was funded by the ISTC poject #4. Thank you fo attention 4// 8th US/ussian Space Suveillance Wokshop,

Generalized functions and statistical problems of orbital mechanics

Generalized functions and statistical problems of orbital mechanics Genealized functions and statistical poblems of obital mechanics S.A. Meshcheyakov. SNIIMASH, 47, Moscow egion, Koolev, Pioneskaya 4 phone: (495) 5 486 E-mail: SMeshcheyakov@mtu-net.u,. Intoduction A new