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Detection of Unknown Space Objects Based on Optimal Sensor Tasking and Hypothesis Surfaces Using Variational Equations

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Abstract

With the rapid increase in the number of space missions, the hazard of growing space debris population in important space territories like the geosynchronous Earth orbit (GEO) region deserves deeper scrutiny. One of the main challenges is to find optimal sensor tasking schemes to detect objects that are not previously cataloged. This paper introduces a method in which a so-called hypothesis surface is used to indicate valuable sensor viewing directions. The hypothesis surface is created using analytic orbit perturbation theory with crucial additions to capture resonance effects and offer sufficient accuracy for the near-Earth region. So-called hypothesis objects are analytically propagated from an early epoch to an epoch near the observation session. Their average positions over the next 24 h are used to design the hypothesis surface. Two cases are shown: (i) a validation case, where we use the hypothesis surface for detecting GEO objects currently in the two-line element set (TLE) catalog, and (ii) a survey strategy to detect high area-to-mass ratio (HAMR) objects. For a variety of observation strategies, for the validation case, we demonstrate that the hypothesis surface is successfully able to detect a large number of GEO objects.

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, West Virginia University, 1306 Evansdale Drive, Morgantown, WV, 26505, USA

    Smriti Nandan Paul

  2. Air Force Institute of Technology, 2950 Hobson Way, Wright-Patterson AFB, OH, 45433, USA

    Bryan D. Little

  3. School of Aeronautics and Astronautics, Purdue University, 701 West Stadium Avenue, West Lafayette, IN, 47907, USA

    Carolin Frueh

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  1. Smriti Nandan Paul
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Appendix A Analytic Expressions for the Entries of Second Hybrid Moment of a Partially Wrapped Normal Distribution

Appendix A Analytic Expressions for the Entries of Second Hybrid Moment of a Partially Wrapped Normal Distribution

$$\begin{aligned}&\varvec{m}_2 (1,1)= \frac{a}{2}[1-\exp {(-C_{11})}\cos {2\mu _1}] \end{aligned}$$
(A1)

where \(a=1-exp{(-C_{11})}\).

$$\begin{aligned}&\varvec{m}_2 (1,2)=-\frac{a}{2}exp(-C_{11})\sin {2\mu _1} \end{aligned}$$
(A2)
$$\begin{aligned}&\varvec{m}_2 (1,3)= \frac{1}{2}exp(-\frac{1}{2}(C_{11}+2C_{12}+C_{22}))\cos {(\mu _1+\mu _2)}\nonumber \\&+\frac{1}{2}exp(-\frac{1}{2}(C_{11}-2C_{12}+C_{22}))\cos {(\mu _1-\mu _2)}\nonumber \\&-\cos {\mu _1}exp(-C_{11}/2)\cos {\mu _2}exp(-C_{22}/2) \end{aligned}$$
(A3)
$$\begin{aligned}\varvec{m}_2 (1,4)&=\frac{1}{2} exp(-\frac{1}{2}(C_{11}+2C_{12}+C_{22}))\sin {(\mu _1+\mu _2)}\nonumber \\&\quad-\frac{1}{2}exp(-\frac{1}{2}(C_{11}-2C_{12}+C_{22}))\sin {(\mu _1-\mu _2)}\nonumber \\&\quad-\cos {\mu _1}exp(-C_{11}/2)\sin {\mu _2}exp(-C_{22}/2) \end{aligned}$$
(A4)
$$\begin{aligned}\varvec{m}_2 (1,5)&= \frac{1}{2}exp(-\frac{1}{2}(C_{11}+2C_{13}+C_{33}))\cos {(\mu _1+\mu _3)}\nonumber \\&\quad+\frac{1}{2}exp(-\frac{1}{2}(C_{11}-2C_{13}+C_{33}))\cos {(\mu _1-\mu _3)}\nonumber \\&\quad-\cos {\mu _1}exp(-C_{11}/2)\cos {\mu _3}exp(-C_{33}/2) \end{aligned}$$
(A5)
$$\begin{aligned}&\varvec{m}_2 (1,6)=\frac{1}{2} exp(-\frac{1}{2}(C_{11}+2C_{13}+C_{33}))\sin {(\mu _1+\mu _3)}\nonumber \\&-\frac{1}{2}exp(-\frac{1}{2}(C_{11}-2C_{13}+C_{33}))\sin {(\mu _1-\mu _3)}\nonumber \\&-\cos {\mu _1}exp(-C_{11}/2)\sin {\mu _3}exp(-C_{33}/2) \end{aligned}$$
(A6)
$$\begin{aligned}&\varvec{m}_2 (1,7)= -exp\bigg (-\frac{C_{11}}{2}\bigg ) C_{14} \sin {\mu _1} \end{aligned}$$
(A7)
$$\begin{aligned}&\varvec{m}_2 (1,8)=-exp\bigg (-\frac{C_{11}}{2}\bigg ) C_{15} \sin {\mu _1} \end{aligned}$$
(A8)
$$\begin{aligned}&\varvec{m}_2 (1,9)=-exp\bigg (-\frac{C_{11}}{2}\bigg ) C_{16} \sin {\mu _1} \end{aligned}$$
(A9)
$$\begin{aligned}&\varvec{m}_2 (2,2)= \frac{a}{2}\bigg (1+exp(-C_{11})\cos {(2\mu _1)}\bigg ) \end{aligned}$$
(A10)
$$\begin{aligned}\varvec{m}_2 (2,3)&=\frac{1}{2} exp(-\frac{1}{2}(C_{11}+2C_{12}+C_{22}))\sin {(\mu _1+\mu _2)}\nonumber \\&\quad-\frac{1}{2}exp(-\frac{1}{2}(C_{11}-2C_{12}+C_{22}))\sin {(\mu _2-\mu _1)}\nonumber \\&\quad-\cos {\mu _2}exp(-C_{22}/2)\sin {\mu _1}exp(-C_{11}/2) \end{aligned}$$
(A11)
$$\begin{aligned}&\varvec{m}_2 (2,4)= \frac{1}{2}exp(-\frac{1}{2}(C_{11}-2C_{12}+C_{22}))\cos {(\mu _1-\mu _2)}\nonumber \\&-\frac{1}{2}exp(-\frac{1}{2}(C_{11}+2C_{12}+C_{22}))\cos {(\mu _1+\mu _2)}\nonumber \\&-\sin {\mu _1}exp(-C_{11}/2)\sin {\mu _2}exp(-C_{22}/2) \end{aligned}$$
(A12)
$$\begin{aligned}&\varvec{m}_2 (2,5)=\frac{1}{2} exp(-\frac{1}{2}(C_{33}+2C_{13}+C_{11}))\sin {(\mu _1+\mu _3)}\nonumber \\&-\frac{1}{2}exp(-\frac{1}{2}(C_{33}-2C_{13}+C_{11}))\sin {(\mu _3-\mu _1)}\nonumber \\&-\cos {\mu _3}exp(-C_{33}/2)\sin {\mu _1}exp(-C_{11}/2) \end{aligned}$$
(A13)
$$\begin{aligned}&\varvec{m}_2 (2,6)= \frac{1}{2}exp(-\frac{1}{2}(C_{11}-2C_{13}+C_{33}))\cos {(\mu _1-\mu _3)}\nonumber \\&-\frac{1}{2}exp(-\frac{1}{2}(C_{11}+2C_{13}+C_{33}))\cos {(\mu _1+\mu _3)}\nonumber \\&-\sin {\mu _1}exp(-C_{11}/2)\sin {\mu _3}exp(-C_{33}/2) \end{aligned}$$
(A14)
$$\begin{aligned}&\varvec{m}_2 (2,7)= exp(-C_{11}/2)C_{14}\cos {\mu _1} \end{aligned}$$
(A15)
$$\begin{aligned}&\varvec{m}_2 (2,8)= exp(-C_{11}/2)C_{15}\cos {\mu _1} \end{aligned}$$
(A16)
$$\begin{aligned}&\varvec{m}_2 (2,9)= exp(-C_{11}/2)C_{16}\cos {\mu _1} \end{aligned}$$
(A17)
$$\begin{aligned}&\varvec{m}_2 (3,3)= \frac{b}{2}[1-\exp {(-c_{22})}\cos {2\mu _2}] \end{aligned}$$
(A18)

where \(b=1-exp{(-C_{22})}\).

$$\begin{aligned}&\varvec{m}_2 (3,4)=-\frac{b}{2}exp(-C_{22})\sin {2\mu _2} \end{aligned}$$
(A19)
$$\begin{aligned}&\varvec{m}_2 (3,5)= \frac{1}{2}exp(-\frac{1}{2}(C_{22}+2C_{23}+C_{33}))\cos {(\mu _2+\mu _3)}\nonumber \\&+\frac{1}{2}exp(-\frac{1}{2}(C_{22}-2C_{23}+C_{33}))\cos {(\mu _2-\mu _3)}\nonumber \\&-\cos {\mu _2}exp(-C_{22}/2)\cos {\mu _3}exp(-C_{33}/2) \end{aligned}$$
(A20)
$$\begin{aligned}&\varvec{m}_2 (3,6)=\frac{1}{2} exp(-\frac{1}{2}(C_{22}+2C_{23}+C_{33}))\sin {(\mu _2+\mu _3)}\nonumber \\&-\frac{1}{2}exp(-\frac{1}{2}(C_{22}-2C_{23}+C_{33}))\sin {(\mu _2-\mu _3)}\nonumber \\&-\cos {\mu _2}exp(-C_{22}/2)\sin {\mu _3}exp(-C_{33}/2) \end{aligned}$$
(A21)
$$\begin{aligned}&\varvec{m}_2 (3,7)= -exp\bigg (-\frac{C_{22}}{2}\bigg ) C_{24} \sin {\mu _2} \end{aligned}$$
(A22)
$$\begin{aligned}&\varvec{m}_2 (3,8)= -exp\bigg (-\frac{C_{22}}{2}\bigg ) C_{25} \sin {\mu _2} \end{aligned}$$
(A23)
$$\begin{aligned}&\varvec{m}_2 (3,9)= -exp\bigg (-\frac{C_{22}}{2}\bigg ) C_{26} \sin {\mu _2} \end{aligned}$$
(A24)
$$\begin{aligned}&\varvec{m}_2 (4,4)= \frac{b}{2}\bigg (1+exp(-C_{22})\cos {(2\mu _2)}\bigg ) \end{aligned}$$
(A25)
$$\begin{aligned}\varvec{m}_2 (4,5)&=\frac{1}{2} exp(-\frac{1}{2}(C_{33}+2C_{23}+C_{22}))\sin {(\mu _3+\mu _2)}\nonumber \\&\quad-\frac{1}{2}exp(-\frac{1}{2}(C_{33}-2C_{23}+C_{22}))\sin {(\mu _3-\mu _2)}\nonumber \\&\quad-\cos {\mu _3}exp(-C_{33}/2)\sin {\mu _2}exp(-C_{22}/2) \end{aligned}$$
(A26)
$$\begin{aligned}\varvec{m}_2 (4,6)&= \frac{1}{2}exp(-\frac{1}{2}(C_{22}-2C_{23}+C_{33}))\cos {(\mu _2-\mu _3)}\nonumber \\&\quad-\frac{1}{2}exp(-\frac{1}{2}(C_{22}+2C_{23}+C_{33}))\cos {(\mu _2+\mu _3)}\nonumber \\&\quad-\sin {\mu _2}exp(-C_{22}/2)\sin {\mu _3}exp(-C_{33}/2) \end{aligned}$$
(A27)
$$\begin{aligned}&\varvec{m}_2 (4,7)= exp(-C_{22}/2)C_{24}\cos {\mu _2} \end{aligned}$$
(A28)
$$\begin{aligned}&\varvec{m}_2 (4,8)= exp(-C_{22}/2)C_{25}\cos {\mu _2} \end{aligned}$$
(A29)
$$\begin{aligned}&\varvec{m}_2 (4,9)= exp(-C_{22}/2)C_{26}\cos {\mu _2} \end{aligned}$$
(A30)
$$\begin{aligned}&\varvec{m}_2 (5,5)= \frac{c}{2}[1-\exp {(-C_{33})}\cos {2\mu _3}] \end{aligned}$$
(A31)

where \(c=1-exp{(-C_{33})}\).

$$\begin{aligned} \varvec{m}_2 (5,6)= & {} -\frac{c}{2}exp(-C_{33})\sin {2\mu _3} \end{aligned}$$
(A32)
$$\begin{aligned} \varvec{m}_2 (5,7)= & {} -exp\bigg (-\frac{C_{33}}{2}\bigg ) C_{34} \sin {\mu _3} \end{aligned}$$
(A33)
$$\begin{aligned} \varvec{m}_2 (5,8)= & {} -exp\bigg (-\frac{C_{33}}{2}\bigg ) C_{35} \sin {\mu _3} \end{aligned}$$
(A34)
$$\begin{aligned} \varvec{m}_2 (5,9)= & {} -exp\bigg (-\frac{C_{33}}{2}\bigg ) C_{36} \sin {\mu _3} \end{aligned}$$
(A35)
$$\begin{aligned} \varvec{m}_2 (6,6)= & {} \frac{c}{2}\bigg (1+exp(-C_{33})\cos {(2\mu _3)}\bigg ) \end{aligned}$$
(A36)
$$\begin{aligned} \varvec{m}_2 (6,7)= & {} exp(-C_{33}/2)C_{34}\cos {\mu _3} \end{aligned}$$
(A37)
$$\begin{aligned} \varvec{m}_2 (6,8)= & {} exp(-C_{33}/2)C_{35}\cos {\mu _3} \end{aligned}$$
(A38)
$$\begin{aligned} \varvec{m}_2 (6,9)= & {} exp(-C_{33}/2)C_{36}\cos {\mu _3} \end{aligned}$$
(A39)
$$\begin{aligned} \varvec{m}_2 (7,7)= & {} C_{44} \end{aligned}$$
(A40)
$$\begin{aligned} \varvec{m}_2 (7,8)= & {} C_{45} \end{aligned}$$
(A41)
$$\begin{aligned} \varvec{m}_2 (7,9)= & {} C_{46} \end{aligned}$$
(A42)
$$\begin{aligned} \varvec{m}_2 (8,8)= & {} C_{55} \end{aligned}$$
(A43)
$$\begin{aligned} \varvec{m}_2 (8,9)= & {} C_{56} \end{aligned}$$
(A44)
$$\begin{aligned} \varvec{m}_2 (9,9)= & {} C_{66} \end{aligned}$$
(A45)

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Paul, S.N., Little, B.D. & Frueh, C. Detection of Unknown Space Objects Based on Optimal Sensor Tasking and Hypothesis Surfaces Using Variational Equations. J Astronaut Sci 69, 1179–1215 (2022). https://doi.org/10.1007/s40295-022-00333-z

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