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Research Papers

Improved Perturbative Solution of Yaroshevskii's Planetary Entry Equation

[+] Author and Article Information
Sarag J. Saikia

School of Aeronautics and Astronautics,
Purdue University,
701 West Stadium Avenue,
West Lafayette, IN 47907-2045
e-mail: sarag@purdue.edu

Jeffrey F. Rhoads

Fellow ASME
Associate Professor
School of Mechanical Engineering,
Purdue University,
585 Purdue Mall,
West Lafayette, IN 47907-2088
e-mail: jfrhoads@purdue.edu

James M. Longuski

Professor
School of Aeronautics and Astronautics,
Purdue University,
701 West Stadium Avenue,
West Lafayette, IN 47907-2045
e-mail: longuski@purdue.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 24, 2014; final manuscript received April 18, 2016; published online June 7, 2016. Assoc. Editor: Haiyan Hu.

J. Comput. Nonlinear Dynam 11(5), 051026 (Jun 07, 2016) (17 pages) Paper No: CND-14-1331; doi: 10.1115/1.4033553 History: Received December 24, 2014; Revised April 18, 2016

An improved approximate analytical solution is developed for Yaroshevskii's classical planetary entry equation for the ballistic entry of a spacecraft into planetary atmospheres at circular speed. Poincaré's method of small parameters is used to solve for the altitude and flight path angle as a function of the spacecraft's speed. From this solution, other important expressions are developed including deceleration, stagnation-point heat rate, and stagnation-point integrated heat load. The accuracy of the solution is assessed via numerical integration of the exact equations of motion. The solution is also compared to the classical solutions of Yaroshevskii and Allen and Eggers. The new second-order analytical solution is more accurate than Yaroshevskii's fifth-order solution for a range of shallow (−3 deg) to steep (up to −90 deg) entry flight path angles, thereby extending the range of applicability of the solution as compared to the classical Yaroshevskii solution, which is restricted to an entry flight path of approximately −40 deg.

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References

Figures

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Fig. 1

Coordinate systems for the three-degree-of-freedom motion of a spacecraft inside a planet's atmosphere. Adapted from Ref. [12].

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Fig. 2

Comparison of classical Yaroshevskii's, new analytical, and numerical solutions of Yaroshevskii's equation: (a) initial conditions in Eq. (94) and (b) initial conditions in Eq. (95)

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Fig. 3

Comparison of percentage relative errors of the classical Yaroshevskii's and the new analytical solutions: (a) initial conditions in Eq. (94) and (b) initial conditions in Eq. (95)

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Fig. 4

Comparison of analytical and numerical altitude solutions: (a) γ0=−10 deg and (b) γ0=−70 deg

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Fig. 5

Comparison of relative error of analytical altitude solutions: (a) γ0=−10 deg and (b) γ0=−70 deg

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Fig. 6

Comparison of analytical and numerical flight path angle solutions: (a) γ0=−10 deg and (b) γ0=−70 deg

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Fig. 7

Comparison of relative error of analytical flight path angle solutions: (a) γ0=−10 deg (b) γ0=−70 deg

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Fig. 8

Comparison of analytical, classical Yaroshevskii's, and numerical altitude solutions: (a) γ0=−10 deg (b) γ0=−70 deg

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Fig. 9

Comparison of percentage relative errors of the altitude solutions: (a) γ0=−10 deg (b) γ0=−70 deg

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Fig. 10

Comparison of percentage relative errors of the altitude solutions: (a) γ0=−10 deg and (b) γ0=−70 deg

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Fig. 11

Comparison of percentage relative errors of the flight path angle solutions: (a) γ0=−10 deg and (b) γ0=−70 deg

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Fig. 12

Comparison of analytical, classical Yaroshevskii's, and numerical deceleration solutions: (a) γ0=−10 deg and (b) γ0=−70 deg

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Fig. 13

Comparison of analytical, classical Yaroshevskii's, and numerical stagnation-point heat rate solutions: (a) γ0=−10 deg and (b) γ0=−70 deg

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Fig. 14

Comparison of analytical, classical Yaroshevskii's, and numerical stagnation-point heat rate solutions: (a) γ0=−10 deg and (b) γ0=−70 deg

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Fig. 15

Comparison of analytical, classical Allen–Eggers, and numerical altitude solutions: (a) γ0=−5 deg (b) γ0=−70 deg

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Fig. 16

Comparison of percentage relative errors of the altitude solutions: (a) γ0=−5 deg (b) γ0=−70 deg

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Fig. 17

Comparison of analytical, classical Allen–Eggers, and numerical flight path angle solutions: (a) γ0=−5 deg and (b) γ0=−70 deg

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Fig. 18

Comparison of percentage relative errors of the flight path angle solutions: (a) γ0=−5 deg and (b) γ0=−70 deg

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Fig. 19

Comparison of analytical, classical Allen–Eggers, and numerical deceleration solutions: (a) γ0=−5 deg and (b) γ0=−70 deg

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