APPLICATION OF A DIRAC DELTA DIS-INTEGRATION TECHNIQUE TO THE STATISTICS OF ORBITING OBJECTS

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1 APPLICATION OF A DIRAC DELTA DIS-INTEGRATION TECHNIQUE TO THE STATISTICS OF ORBITING OBJECTS Dario Izzo Advanced Concepts Team, ESTEC, AG Noordwijk, Te Neterlands ABSTRACT In many problems related to te description of te debris environment it is mandatory to be able to deal wit orbital dynamics from a statistical point of view. Tis allows one to compute debris spatial densities, to evaluate collision probabilities and to improve te processing of te data received from observational campaigns. In tis paper we use a novel tecnique based on te matematical manipulation of Dirac s delta functions and we sow ow tis is a powerful new tool in dealing wit orbital mecanics from a statistical point of view. Te metodology is applied to derive some analytical expressions for te density functions associated wit te velocity distribution of orbiting objects. Key words: Statistical Space Fligt Mecanics; Diffused Satellite; Satellites Velocity Distribution; Uncertainties on Orbital Parameters.. INTRODUCTION In tis paper a novel metodology will be used to derive some expressions describing te probability density functions associated wit te velocity distribution of orbiting objects. Te objects are described by te statistical properties of teir orbital parameters troug probability distribution functions. Hypoteses of uncorrelated variables will be used but is not necessary. Te work complements te already publised results [Izzo 5] on te effect of orbital parameter uncertainties wit new expressions on te velocities and some teoretical clarifications.. METHODOLOGY To sow te motivation and te benefits of te novel metodology used trougout tis paper, we start, following Au & Tam 999, by considering a discrete random variable Z wit a probability distribution P given by te following: Z 5 P We ten introduce a one-to-one transformation Y = Z +, and a double-valued transformation Y = Z. It is immediately realized tat te probability distribution for te new variables can be obtained by te following table: Y = Z Y = Z P only in te case of te one-to-one transformation. For te oter case some rearrangement is needed as P Y = 9 = P X = + P X =. Te same problem exists for continuous random variables. For tis reason te standard cange of variable tecnique for probability density functions involves only monotonic transformations and a number of new variables equal to te number of old variables. Tese two fundamental sortcomings of te standard procedure may be solved by te introduction of a novel tecnique introduced independently, and in different contexts, by Au & Tam 999 and Izzo. Te tecnique makes use of some fundamental properties of Dirac s delta functions, and in particular of te following teorem: δ[x ˆxy] = i δ[y y i ] dˆx dy y=y i were y i are all te solutions in y of te equation x = ˆxy. Te teorem as also a multivariate form: δ[x ˆxy] = i δ[y y i ] J y=yi were J = ˆxi y j is te Jacobian of te transformation ˆxy and y i are te solutions to te equation x = ˆxy. In te metodology used ere Dirac s delta functions are used to define an initial probability density function associated wit a deterministic process. Te desired variables are ten dis-integrated from te process and te integral is easily evaluated by exploiting te quoted property of Dirac s delta functions. Te term dis-integration refers ere to a standard procedure in te teory of probability tat is also called randomisation.

2 Let us consider for example te simple linear system: ẋ = Ax wit initial conditions x = x. It is well known tat its solution may be defined, by introducing te matrix exponential, as ˆxx, t = e At x. In tis deterministic situation, were all te information needed to predict te evolution of te system is known, te probability density function associated wit te state x may be written: ρx x, t = δ[x ˆxx, t] We may now tink of te initial condition x as a random variable wit probability distribution function fx and we may terefore dis-integrate it by writing: ρx t = δ[x ˆxx, t]fx dx x D were D is te domain of f. By making use of te multivariate version of te teorem on Dirac s Delta variable cange stated by Eq., it is possible to write: ρx t = δ[x x ]fx e At dx x D were x is te only solution of equation x = ˆxx, t given by x = e At x. We are terefore able to write a final expression for te probability density function associated wit te variable x wenever te initial conditions of te dynamical system tat describe te time evolution of x are considered random and ave to be dis-integrated from te process: ρx t = feat x e At Note tat in tis simple case te same expression could be reaced by using a more standard tecnique of cange of variable. Te transformation x = e At x from x to x is in fact a one-to-one transformation tis is always te case for dynamical systems for wic te Caucy teorem olds and te new variables x are as many as te old ones x. Te metodology outlined does not suffer from tese limitations and can terefore be applied to general cases, in particular to cases in wic we wis to consider time as random, and terefore also te most simple generic linear system considered above does not admit a oneto-one transformation.. APPLICATIONS TO SPACE FLIGHT MECHANICS description of te space debris environment in particular, by Izzo ; Izzo & Valente 4; Izzo 5. Te spatial density of a large family of orbiting objects as been derived analytically for Geostationary, LEO and Molnyia satellites, extending known results coming from past approaces [Izzo 5]. More results are derived ere to sow te use of Dirac s delta wen dis-integrating random variables from a given stocastical process, and in particular in dealing wit uncertainties related to te orbital parameters and teir effects on velocity distributions... Te odograp plane In tis section we briefly recall standard results on te velocity vector along a Keplerian orbit. In particular we start from te definitions of two important motion invariants: = r v µ e = v µ r r tat is te angular momentum vector and te Laplace vector. By taking te vector product between tese two quantities we easily get: µ v = e sin νî ρ + + e cos νî θ were ν is te true anomaly along te orbit. Te above expression describes te relation between te velocity and te true anomaly along a Keplerian orbit tat will be used in te following. For an orbit immersed in tree dimensional space it is common to introduce te angle θ = ν + ω tat will later be used in our calculations... Velocity magnitude distribution Let us consider a satellite on a Keplerian orbit and te probability density function associated wit te variable k = v. Following te metodology outlined above, first a deterministic process is considered: ρk α, t = δ k [k ˆk α, t] were α is a vector containing te initial conditions defining te Keplerian orbit any set of orbital parameters migt be considered and ˆk α, t = µ { p [ + e cos[θt ω]] a }. Ten te time t is considered as random and uniformly distributed over one orbital period so tat te variable is disintegrated from te process: Te metodology described in te previous section as already been applied to different problems related to orbital mecanics in general, and to te ρk α = T δ k [k ˆk α, t]

3 To evaluate te above integral we must cange te variable in te Dirac delta function and we do tis by applying te fundamental property of Dirac s delta functions stated in Eq.: ρk α = T δ t [t t i ] dˆk 4 were we must sum all te solutions in t to te equation ˆk α, t = k. Simple astrodynamics tell us tat tese solutions are given by i cos[θt i ω] = k v a v p v p v a were v a and v p are te apogee and perigee velocities related to te semi-major axis and eccentricity by te relations: e = v p v a, a = µ v p + v a v a v p Two of te above solutions will be contained witin te integration interval of one period, giving nonzero terms in equation 4. Corresponding to tese two solutions we also ave: sin[θt i ω] = ± vp va k vav p k and r = µ k + v av p so tat we may evaluate te derivatives: dˆk = µ p e sin[θt i ω] θt i = µ µ p e sin[θt i ω]b a =... = k + v av p µv a + v p r k v av p k 5 Plugging tis expression into eq.4 we get te final expression: ρk v a, v p = v p + v a vpv a π k + v a v p k vav p k Tis is plotted in Figure for an Eart orbit... Radial velocity distribution We sow ere te calculation tat leads to determining te analytical expression for te probability density function associated wit te variable v ρ, i.e. ρk v a,v p k m /s Figure. Probability density function associated wit te variable k = v on an Eart orbit aving e =.4 and a = km. te radial velocity along an orbit, if te time is considered as random and uniformly distributed witin an orbital period. We start from a deterministic process by writing: ρv ρ α, t = δ v [v ρ ˆv ρ α, t] were ˆv ρ α, t = µe sin[θt ω]. We ten consider time as random wit a uniform probability distribution function, and we dis-integrate it from te process by writing: ρv ρ α = T δ v [v ρ ˆv ρ α, t] We ten make use of te teorem on Dirac s delta functions stated by Eq.: ρv ρ α = T δ t [t t i ] dˆvρ were t i are te solutions in t to te equation ˆv ρ α, t = v ρ given by te expression: sin[θt i ω] = i v ρ were we introduced te maximum value of te radial velocity = µe. We also ave: dˆv ρ = cos[θt i ω] θ µ = cos[θt i ω]b a =... = π T vρ M e r ± e v ρ M v ρ 6 Note tat te values of te derivative are different for te two different time instants t i contained in one orbital period. After some more manipulations we get:

4 ρv ρ, e = vρ M e vρ M + e vρ M π v ρm v ρm e vρ M Tis is plotted in Figure for an Eart orbit Tis way, toug, we would be forced to evaluate te determinant of te Jacobian matrix. By applying eq. two times in cascade we actually save some calculations. We start terefore by eliminating te Dirac delta in v ρ. Taking advantage of te calculations already done in te previous paragrap we get: ρv ρ, v θ = ρv ρ v ρm,e.5..5 = e feδ[v θ ˆv θ ] π + vρ M + e vρ M v ρ m/s Figure. Probability density function associated wit te radial velocity on an Eart orbit aving e =.4 and a = km..4. Radial-Tangential velocity distribution Having acquired confidence wit te new metodology and wit its application to space fligt mecanics problems, we are now ready to build more complicated probability distribution functions. In problems related to collisions between objects belonging to different families of orbits, it is important to be able to describe someow te relative velocity between objects at te time of possible impact. Many oter problems require to be able to write te probability density function associated wit an object velocity. We ere derive a first result on tis issue. In a deterministic process te probability density function associated wit te variables v ρ and v θ may be written as: were ρv ρ, v θ α, t = δ[v ρ ˆv ρ α, t]δ[v θ ˆv θ α, t] ˆv ρ = sin[θt ω] { ˆv θ = v ρm e + cos[θt ω]} We now consider te time as a random variable uniformly distributed in one orbital period, and te eccentricity as a random variable wit wic we associate a probability distribution function fe. By dis-integrating tese two variables we get: ρv ρ, v θ = T ρv ρ, v θ α, tfede We could now use te multivariate form of te Dirac delta property on variable cange stated by eq.. + e feδ[v θ ˆv θ ] π de vρ M e vρ M were ˆv θ,, e = vρ M e ± v ρ M v ρ. We now apply eq. again to eliminate te two Dirac deltas in v θ. By taking into account tat: dˆv θ, de = v ρ M e 7 and tat te two solutions in e to te two equations v θ = ˆv θ, are: e, = v θ vρ M we get a final expression of te form: ρv ρ, v θ = = π v θ vρ M i= e i fe i 8 Te expression above still contains as a conditioning variable. Tis variable migt also be disintegrated leading to a formula solvable by quadrature..5. One last example As a last example we ere derive a quite remarkable formula for te probability distribution function associated wit te variables r, v ρ, v θ. ρr, v ρ, v θ α, t = δ[r ˆr α, t]δ[v ρ ˆv ρ α, t]δ[v θ ˆv θ α, t] Following te usual metodology, we first consider te time as random and uniformly distributed in one

5 period, and ten te eccentricity as random and described by a probability density function f. We get te expression: ρr, v ρ, v θ = i= π v θ v ρ M v ρ e i fe i δ[r ˆr i v θ, ] 9 were ˆr i v θ, = µe i v θ. We now take one last step by considering as random and wit a probability distribution function g and we eliminate te last Dirac delta by dis-integrating te random variable from te orbital process. By taking into account tat: dˆr i d = µ vρ M vρ M and tat te only solution in of te equation r = ˆr i is: vρ µ M = µr r vρ written in terms of Dirac s delta proves to be very useful for deriving te velocity distributions of a family of objects. Some new formulas ave been derived ere tat te autor opes will prove to be useful tools improving our understanding of te debris environment and of all tose situations in wic orbiting objects ave to be described by a probabilistic distribution rater tan by a certain position and velocity. REFERENCES Au C., Tam J., 999, American Statistician, 5, 7 Izzo D.,, Statistical Modelling of Sets of Orbits, P.D. tesis, Department of Matematical Metdods and Models, University La Sapienza, Rome. Available on-line. Izzo D., 5, Journal of Guidance Control and Dynamic,, 54 Izzo D., Valente C., 4, Acta Astronautica, 54, 54 we get, applying again te Dirac delta property stated in eq. and after some basic algebraic manipulations, a final expression in te form: ρr, v ρ, v θ = = µ π v θ gvρ M µr r vρ i= e i fe i Tis probability density function is no longer conditioned by te knowledge of any initial condition. We ave managed to describe te velocity distribution at a certain distance of a satellite, or a group of objects, wose uncertain orbits are described via te density functions associated wit two orbital parameters, te eccentricity e and te maximum radial velocity ρ M. Te above equation is valid under te ypotesis tat tese two random variables are uncorrelated, but a similar expression may also be obtained if a joint probability distribution function is available. 4. CONCLUSIONS Te tecnique based on te dis-integration of te orbital parameters from an initial density function