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A Wave Scattering Approach to Modelling Surface Roughness in Orbital Aerodynamics

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Surface roughness is the missing variable in high-altitude satellite drag models.

desk verdict A genuinely new analytical roughness kernel for gas-surface interaction, thoroughly verified against TPMC, but the satellite-drag attribution rests on fitting the very data it claims to explain. read the letter →

arxiv 2411.10874 v1 pith:WUUTTSTD submitted 2024-11-16 astro-ph.EP astro-ph.IMphysics.space-ph

Sabin Viorel Anton , Bernardo Sousa Alves , Christian Siemes , Jose van den IJssel , Pieter N.A.M. Visser This is my paper · ORCID
classification astro-ph.EPastro-ph.IMphysics.space-ph
keywords gas-surfaceinteractionorbitalaerodynamicssurfaceroughnesswavescatteringKirchhoffapproximationpoly-Gaussiansurfacessatellitedragfreemolecularflow
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Orbital aerodynamics in low Earth orbit is usually computed with gas-surface interaction models that assume smooth surfaces and one or two empirical accommodation parameters, and those models mismatch tracking data for spherical satellites above about 400 km. The paper argues that the missing ingredient is geometric surface roughness, which produces shadowing, multiple reflections, and backscattering of incoming gas particles. It supports this with a physics-based scattering kernel derived from wave scattering theory, in which a gas particle's de Broglie wave obeys a Helmholtz equation with a statistically rough rigid boundary. The kernel reproduces test-particle Monte Carlo simulations and experimental Argon and Helium scattering from smooth and rough Kapton and Aluminium, and it yields altitude-dependent drag coefficients matching the Stella and Gridsphere satellites when the roughness parameter $\sigma/R$ is near 0.55 and 0.85. If the paper is right, surface roughness is a first-order variable in thermospheric drag, not a correction to be folded into empirical accommodation coefficients.

What carries the argument

The central object is the Kirchhoff wave-scattering kernel, an analytic expression for the angular distribution of gas particles reflected from a statistically rough surface, obtained by solving the Helmholtz equation for the particle's de Broglie wave with a rigid boundary condition. It is parameterized by a poly-Gaussian surface model—the height profile is a Gaussian mixture with coefficients $\sigma_k$ and $\mu_k$ and correlation length $R$—and by a local scattering kernel for atomic-scale interactions. The kernel carries the argument by turning surface height statistics into scattering statistics, while a Smith-type shadowing function and an iterative sampling algorithm extend it to multiple collisions and backscattering. In its Gaussian limit the whole roughness effect reduces to a single parameter $\sigma/R$, which is sufficient for the spherical-satellite comparisons.

What would settle it

Track a spherical satellite whose surface roughness has been independently measured, compute its drag coefficient with the Gaussian Kirchhoff kernel using the measured $\sigma/R$, and compare with accelerometer- or tracking-derived drag over 400 to 1000 km; if the altitude trend deviates beyond the model's stated error, the roughness explanation fails.

Watch

Extended reading notes

Core claim

The central claim is that a gas particle scattering off a real satellite surface can be treated as a wave scattering from a rough interface, and that the macroscopic consequences of roughness—shadowing, multiple reflections, and backscattering—quantitatively explain observations that smooth-surface kernels cannot. The paper derives a closed-form Kirchhoff scattering kernel for the probability density of reflected directions as a function of incidence angle, poly-Gaussian surface statistics $\sigma_k$ and $\mu_k$, and autocorrelation length $R$, together with an analytic shadowing function and an iterative multi-reflection algorithm. This kernel is wrapped around an arbitrary local scattering model for the atomic-scale interaction. The combined model is verified against ray-tracing Monte Carlo simulations across the local parameter space, reproduces measured scattering of noble gases from smooth and eroded Kapton and aluminium, and, applied to a sphere, reproduces the drag-coefficient altitude profiles of the Stella and Gridsphere satellites with $\sigma/R = 0.55$ and $0.85$ where the standard diffuse and quasi-specular kernels diverge from tracking data.

Load-bearing premise

The load-bearing premise is that successive surface collisions are statistically independent and that height and slope distributions are independent; the paper itself reports up to 7% error when this fails for low tangential momentum accommodation with high normal accommodation.

Editorial extensions

If this is right

  • Surface roughness raises the drag coefficient of a sphere in the helium-dominated thermosphere above 400 km, with Stella and Gridsphere data reproduced at $\sigma/R = 0.55$ and $0.85$.
  • At high roughness the new kernel's drag coefficient approaches that of the diffuse DRIA model, giving a physical explanation for DRIA's empirical success at lower altitudes.
  • Backscattering at near-parallel incidence increases drag on flow-exposed angled surfaces beyond what quasi-specular smooth-surface kernels predict.
  • The one-parameter Gaussian version of the model is sufficient for drag estimation, meaning the roughness parameter can be fitted from orbital acceleration data in the same way existing empirical parameters are fitted.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable consequence not explored in the paper: the same roughness parameters could be measured pre-flight from power spectral densities of engineering surfaces, giving drag predictions that require no in-orbit calibration at all.
  • The paper's interpretation of near-unity tangential accommodation suggests laboratory measurements on rough coupons should be revisited, since part of what is called tangential accommodation may actually be geometric backscattering.
  • If the mechanism generalizes beyond spheres, roughness should alter lift and side forces on attitude-controlled satellites as well as drag, with implications for torque and attitude dynamics that the paper does not compute.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops a gas-surface interaction model for orbital aerodynamics based on Kirchhoff wave scattering from statistically described rough surfaces. It extends a poly-Gaussian surface model to slope statistics, derives a closed-form scattering kernel with a shadowing function and an iterative multi-reflection algorithm (Eqs. 79, 108, 113), verifies the kernel against test-particle Monte Carlo simulations for Gaussian and non-Gaussian surfaces, compares with published Ar/He scattering experiments on Kapton, and applies the model to flat-plate and spherical-satellite drag coefficients. The authors conclude that surface roughness explains previously reported inconsistencies between DRIA/CLL predictions and Stella/Gridsphere tracking data.

Significance. The model is a potentially significant methodological contribution: it provides a closed-form, physics-inspired kernel that captures backscattering, shadowing, and multi-reflections, and it passes a meaningful TPMC closure test, with errors below 2% for Gaussian surfaces and up to 7% for a worst-case poly-Gaussian surface, errors that the authors correctly attribute to independence assumptions in the shadowing function. The TPMC error maps and the qualitative reproduction of observed backscattering are strengths. However, the headline application to spherical satellites is not an independent test: the only roughness parameter is fitted to the very Stella/Gridsphere observations the paper claims to explain. The causal claim therefore needs to be reframed or supported by independent roughness characterization.

major comments (3)
  1. [Sec. 4.3, Figs. 24-25, Table 6] The central causal claim that surface roughness explains the Stella/Gridsphere discrepancies is not supported by the analysis as presented. The text states that sigma/R = 0.55 and 0.85 were 'optimised to fit the observations' of these two satellites, and no independent measurement of the surface PSD, autocorrelation length, or local accommodation parameters is used; the route advertised in Eq. (83) (ground PSD + MD) is not followed. Figures 24-25 therefore demonstrate the existence of a two-value fit rather than that roughness is the physical cause, because errors in the local kernel, isotherm parameters, or atmospheric model could be absorbed into the fitted sigma/R values. The abstract and Section 4.3 should be reframed, or an out-of-sample test with independently characterized roughness should be added.
  2. [Sec. 4.2, Table 3] The Kapton validation is qualitative. The local CLL parameters alpha_N, sigma_T and the physisorption fraction are determined by trial-and-error, and the poly-Gaussian transformations mu(gamma), sigma(gamma) are chosen to visually match electron microscope images, so the agreement in Fig. 21 is to some extent a demonstration of the model's expressiveness rather than an independent validation. The authors should state this limitation explicitly in the validation claim, or add a quantitative goodness-of-fit metric and a sensitivity analysis to the hand-tuned parameters.
  3. [Sec. 3.4, Sec. 4.1, Table 6] The spherical-satellite application sets sigma_T = 0, which is precisely the parameter region the authors themselves identify as the least accurate: the text following Fig. 11 notes large discrepancies for sigma_T = 0.0, alpha_N = 1.0, and Fig. 18 shows up to 7% error in this corner for the poly-Gaussian version. The Gaussian error map in Fig. 29 does not include sigma_T = 0.0, so the kernel's accuracy in the regime actually used for the sphere is unverified. The fitted sigma/R = 0.55 and 0.85 and the resulting CD curves should therefore be treated with caution unless this corner is validated.
minor comments (4)
  1. [Sec. 1] Remove the editorial note 'check the bib file because O. and I. should not appear here' and complete the Aksenova & Khalidov (2008) reference with proper author initials.
  2. [Sec. 4.1] Replace the placeholder '(?)' with a working repository/DOI for the GSI_ToolBox software; the claim of published open-source software is currently not verifiable.
  3. [Eqs. (94)-(95)] The symbol T is used in place of the autocorrelation length R in the shadowing expression; compare with Eq. (35).
  4. [Figs. 24-25] The legends read 'Kr, /T = 0.55' and 'Kr, /T = 0.85'; these should be sigma/R.

Circularity Check

1 steps flagged · score 6.0 of 10

The Stella/Gridsphere roughness attribution is a two-parameter fit to the very data it claims to explain; the kernel derivation and TPMC verification are otherwise self-contained.

  1. fitted input called prediction [Sec. 4.3, sphere-altitude analysis, Figs. 24-25 and Table 6]
    "For the Kirchhoff model, two curves were generated in each figure, corresponding to roughness parameters of σ/R = 0.55 and σ/R = 0.85, which were optimised to fit the observations of the two high-altitude spherical satellites alongside the low-altitude data. ... Comparing the two drag coefficients of the Kirchhoff model with the Stella and Gridsphere data points, it appears that Stella is best modelled by a roughness parameter of σ/R = 0.55 while Gridsphere corresponds to σ/R = 0.85."

    The paper's headline conclusion is that previously observed inconsistencies between DRIA/CLL models and tracking data for spherical satellites can be attributed to surface roughness. In Sec. 4.3 this attribution is supported by setting σ/R = 0.55 and 0.85 so that the model reproduces the Stella and Gridsphere observations themselves, together with low-altitude data. Figs. 24-25 therefore demonstrate the existence of a fit, not a prediction. No independent roughness measurement, surface PSD, or molecular-dynamics parameter determination — the route advertised in the abstract and Eq. (83) — is used. Errors in the local CLL kernel, in the borrowed DRIA isotherm parameters, or in the atmospheric model can be absorbed into the fitted σ/R values.

full rationale

The mathematical core (Sections 3.1-3.6) is self-contained: the Kirchhoff integral and poly-Gaussian surface statistics are derived in the paper, with only standard external results (Beckman, Smith, Brown, Litvak-Malyugin) and the local CLL kernel imported, none of which is the paper's conclusion. The TPMC verification in Sec. 4.1 is an independent numerical benchmark and shows genuine agreement, with acknowledged relative errors up to 7% in the (alpha_N, sigma_T) = (1.0, 0.0) corner. The Kapton comparison in Sec. 4.2 is partially fitted: the local CLL parameters and physisorption fraction are found by trial and error from the smooth-surface scattering data, while the rough-surface mu(gamma) and sigma(gamma) are chosen visually from microscope images; the rough-surface prediction retains some independent content, so this is weaker than the sphere step but not fully circular. The central circular step is the sphere application: the roughness parameters are optimised to fit Stella and Gridsphere data, and the same figures are then presented as showing that roughness explains the prior inconsistency with DRIA and CLL. Because no independent roughness characterisation is used, the causal statement in the abstract reduces to a two-parameter fit. Self-citations to Siemes et al. and March et al. are contextual and not load-bearing. On balance, the paper has a genuinely independent model derivation and numerical verification, but its flagship satellite-attribution claim is partially circular, giving a score of 6.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The model's central claim rests on: (i) a wave-particle mapping with Kirchhoff tangent-plane scattering, (ii) a statistical surface model limited to Gaussian autocorrelation, (iii) independence assumptions in the shadowing function, and (iv) local CLL parameters that are partly fitted to experimental data. The roughness parameters used to explain satellite drag are fitted to the same data, which is the largest circularity burden. No new physical entities are proposed.

free parameters (5)
  • Poly-Gaussian Hermite coefficients sigma_k, mu_k = N up to 40; e.g. Table 2 and Table 3 choices; Gaussian limit reduces to sigma/R
    These define the surface height and slope PDFs that drive the scattering kernel. They are chosen by hand or to visually match microscope images, and are the main surface descriptors.
  • Gaussian roughness parameter sigma/R = 0.2, 0.4, 0.8 in verification; 0.55 and 0.85 for Stella/Gridsphere
    In the Gaussian limit the entire surface model reduces to this single parameter; the satellite values are optimised to fit the Stella/Gridsphere CD data, not measured on-orbit.
  • Local CLL accommodation coefficients alpha_N, sigma_T = Table 3: 0.6, 0.2 for Kapton comparison
    Found by trial and error to match smooth Kapton scattering; the model's predictive content for rough Kapton depends on these fitted local parameters.
  • Physisorption fraction f_physisorption = 0.45
    Chosen by trial and error to reproduce smooth Kapton data; affects the diffuse versus specular split in the Kapton comparison.
  • SESAM isotherm parameters K, B, Xi, zeta = Taken from Walker et al. (2014b), Table 6
    Used to convert atomic oxygen partial pressure into surface coverage; not fitted in this paper but are empirical inputs that influence the sphere CD curves.
assumptions (7)
  • domain assumption Free molecular flow: Knudsen number much larger than 1, intermolecular collisions neglected
    Section 3.1 opening; appropriate for LEO but an assumption.
  • domain assumption Surface is a perfectly rigid infinite step potential; local reflection is specular (Eq. 11)
    Required to map gas particles to the Helmholtz/Kirchhoff problem; local interactions are re-injected through a separate local kernel, which is itself an assumption.
  • standard math Kirchhoff approximation: scattered field on the surface equals that of an infinite tangent plane, valid when 4*pi*R*cos(nu) >> lambda (Eq. 115)
    Borrowed from EM scattering; the paper checks the condition with the He de Broglie wavelength and finds it valid for lattice-scale roughness.
  • ad hoc to paper Surface is isotropic, homogeneous, and representable by a poly-Gaussian process with Gaussian autocorrelation C_epsilon = C_gamma = exp(-r^2/R^2)
    Section 3.2; this limits the surface PSD to a Gaussian form (Eq. 116), excluding fractal surfaces, which the authors acknowledge in Section 3.7.
  • ad hoc to paper Height and slope processes are independent and successive collision points are statistically independent (Eqs. 27 and 90)
    Needed for the closed-form shadowing function; the authors identify this as the likely cause of up to 7% error for sigma_T near 0 and alpha_N near 1.
  • domain assumption CLL kernel accurately describes local interactions on a smooth surface
    Used throughout as the local kernel; if CLL is wrong, the composite model inherits its errors.
  • ad hoc to paper Linearization of mu(x,y) and sigma(x,y) around P0 and neglect of the cross-correlation term 2*gamma_dot_x*gamma_dot_y*Delta_x*Delta_y
    Section 3.3 needed to obtain Eq. (79); not justified beyond small roughness gradients.

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Pith. "Pith review of A Wave Scattering Approach to Modelling Surface Roughness in Orbital Aerodynamics." pith.science (2026). https://pith.science/paper/WUUTTSTD

@misc{pith2026241110874,
  author       = {Pith},
  title        = {Pith review of: A Wave Scattering Approach to Modelling Surface Roughness in Orbital Aerodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WUUTTSTD}},
  note         = {Machine review of arXiv:2411.10874}
}
read the original abstract

The increasing density of space objects in low-Earth orbit highlights the critical need for accurate orbit predictions to minimise operational disruptions. One significant challenge lies in accurately modelling the interaction of gas particles with the surfaces of these objects, as errors in aerodynamic coefficient modelling directly impact orbit prediction accuracy. Current approaches rely on empirical models, such as those by Sentman and Cercignani-Lampis-Lord, incorporating one or two adjustable parameters typically calibrated with orbital acceleration data. However, these models fall short in capturing essential gas-solid interaction processes, including multiple reflections, shadowing, and backscattering caused by surface roughness. We present a novel, physics-based gas-surface interaction model that utilises electromagnetic wave theory to account for macroscopic effects of surface roughness on gas particle scattering distributions. This approach not only offers a more accurate representation of gas-surface interactions but also allows parameter determination through a combination of ground-based surface roughness measurements and molecular dynamics simulations at the atomic scale. The model validity is tested across the entire parameter space using a test-particle Monte Carlo method on a simulated rough surface. Furthermore, it successfully reproduces experimental results from the literature on the scattering of Argon and Helium from smooth and rough Kapton and Aluminium surfaces. Finally, we demonstrate the model's impact on aerodynamic coefficients for simple geometric shapes, comparing the results with those from the Sentman and Cercignani-Lampis-Lord models. This comparison reveals that inconsistencies previously observed between these models and tracking data for spherical satellites can be attributed to surface roughness effects, which our model effectively accounts for.

Figures

Figures reproduced from arXiv: 2411.10874 by the authors.

Figure 1
Figure 1. A sketch of the radially-averaged power spectral density of a typical engineering surface. Three regions are depicted: (left) a macroscopic roughness [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. On the left: a sketch of the geometrically-rough profile of a typical engineering surface, and the multi-scattering of gas particles, with arrow thickness [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. A sketch of the angular distributions of three scattering kernels for gas-surface interaction. The [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (28 more)
Figure 4
Figure 4. Figure 4: A sketch of three different physical gas-surface interaction models. The Hard Cube model (left) assumes rigid, elastic collisions between gas and surface atoms, the latter oscillating in the z-axis with a Maxwellian thermal velocity. The Soft Cube model further assumes…
Figure 5
Figure 5. Figure 5: Left: The power spectral density of a general rough surface, and the scales included in each GSI model. Bottom-right: A realisation of Gaussian geometric [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: A sketch of the coordinate system employed by the Kirchho [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: A sketch of a plane wave reflecting off a rough surface, observed from a point, P, situated infinitely far away from the surface in the direction of kr. Here, R ′ is the distance between the observer and point B, R0 is the distance between the same observer and the ori…
Figure 8
Figure 8. Figure 8: A sketch of a rough surface’s self-shadowing e [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: A flowchart of the proposed GSI model iterative process of accounting for multiple reflections. [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: On the left: illustrations of three Gaussian surface samples with roughness values of 0.2, 0.4 and 0.8. On the right: the height ( [PITH_FULL_IMAGE:figures/full_fig_p029_10.png]
Figure 11
Figure 11. Figure 11: The YZ-plane marginal angular PDFs of Helium gas particles with di [PITH_FULL_IMAGE:figures/full_fig_p031_11.png]
Figure 12
Figure 12. Figure 12: The YZ-plane marginal normal and tangential velocity PDFs of helium gas particles with an incidence angle of [PITH_FULL_IMAGE:figures/full_fig_p032_12.png]
Figure 13
Figure 13. Figure 13: The variation of the global normal and tangential momentum accommodation coe [PITH_FULL_IMAGE:figures/full_fig_p033_13.png]
Figure 14
Figure 14. Figure 14: On the left: illustrations of a Non-Gaussian surface sample. On the right: the height ( [PITH_FULL_IMAGE:figures/full_fig_p034_14.png]
Figure 15
Figure 15. Figure 15: The YZ-plane marginal angular PDFs of helium gas particles with di [PITH_FULL_IMAGE:figures/full_fig_p036_15.png]
Figure 16
Figure 16. Figure 16: The YZ-plane marginal normal and tangential velocity PDFs of helium gas particles with an incidence angle of [PITH_FULL_IMAGE:figures/full_fig_p037_16.png]
Figure 17
Figure 17. Figure 17: The variation of the global normal and tangential momentum accommodation coe [PITH_FULL_IMAGE:figures/full_fig_p038_17.png]
Figure 18
Figure 18. Figure 18: The global momentum relative error between the poly-Gaussian Kirchho [PITH_FULL_IMAGE:figures/full_fig_p038_18.png]
Figure 19
Figure 19. Figure 19: Illustrations of non-Gaussian surfaces resembling atomic oxygen-eroded Kapton (left) and pristine Kapton (right), both generated with the Kirchho [PITH_FULL_IMAGE:figures/full_fig_p040_19.png]
Figure 20
Figure 20. Figure 20: The height (ξ) and slope (ξ˙) PDFs of non-Gaussian surfaces resembling atomic oxygen-eroded Kapton (left) and pristine Kapton (right), both generated with the Kirchhoff model [PITH_FULL_IMAGE:figures/full_fig_p040_20.png]
Figure 21
Figure 21. Figure 21: A comparison between the argon-Kapton scattering results from Erofeev et al. (2012) and the poly-Gaussian Kirchho [PITH_FULL_IMAGE:figures/full_fig_p040_21.png]
Figure 22
Figure 22. Figure 22: The variation of the drag (left) and lift (right) coe [PITH_FULL_IMAGE:figures/full_fig_p042_22.png]
Figure 23
Figure 23. Figure 23: The variation of the drag coefficient of a flat plate with respect to the CLL parameters αN and σT , for different levels of Gaussian roughness and incidence angles. For a final investigation, the Kirchhoff model was used to study the effects of surface roughness on C…
Figure 22
Figure 22. Figure 22: The isotherm parameters for the dependence of the DRIA and CLL [PITH_FULL_IMAGE:figures/full_fig_p044_22.png]
Figure 24
Figure 24. Figure 24: A comparison between the drag coefficient of a sphere computed with the closed forms of the DRIA and CLL kernels as given by Walker et al. (2014a), those generated with the Gaussian Kirchhoff model for different levels of roughness, and fitted drag coefficients of the…
Figure 25
Figure 25. Figure 25: A comparison between the drag coefficient of a sphere computed with the closed forms of the DRIA and CLL kernels as given by Walker et al. (2014a), those generated with the Gaussian Kirchhoff model for different levels of roughness, and fitted drag coefficients of the…
Figure 26
Figure 26. Figure 26: A sketch of a volume Ω, bounded by a closed surface ∂Ω. A point P is shown inside this volume, at the centre of a sphere with radius R. The sphere surface and ∂Ω are connected through a cylindrical channel. Appendix B Additional Results This section presents the angul…
Figure 29
Figure 29. Figure 29: All three analyses use the same simulation parameters, as specified in Table 1 and Table 2. [PITH_FULL_IMAGE:figures/full_fig_p049_29.png]
Figure 27
Figure 27. Figure 27: The XY-plane marginal angular PDFs of Helium gas particles with di [PITH_FULL_IMAGE:figures/full_fig_p050_27.png]
Figure 28
Figure 28. Figure 28: The XY-plane marginal angular PDFs of Helium gas particles with di [PITH_FULL_IMAGE:figures/full_fig_p051_28.png]
Figure 29
Figure 29. Figure 29: The momentum relative error between the Gaussian Kirchho [PITH_FULL_IMAGE:figures/full_fig_p052_29.png]

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Reference graph

Works this paper leans on

85 extracted references · 62 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize "" * " " * ...

  3. [3]

    title Aerodynamic Calculation of Rough Surface in Rarefied Gas Flow Applying the Solution of Inverse Problem

    author Aksenova , & author Khalidov ( year 2008 ). title Aerodynamic Calculation of Rough Surface in Rarefied Gas Flow Applying the Solution of Inverse Problem . In booktitle Proceedings of the 9th European Conference for Aerospace Sciences \/ . :10.13009/EUCASS2022-4897

    work page doi:10.13009/eucass2022-4897 2008

  4. [4]

    , author Miller, S

    author Banks, B. , author Miller, S. , & author de Groh, K. ( year 2004 ). title Low earth orbital atomic oxygen interactions with materials . In booktitle 2nd International Energy Conversion Engineering Conference \/ . publisher American Institute of Aeronautics and Astronautics . :10.2514/6.2004-5638

    work page doi:10.2514/6.2004-5638 2004

  5. [5]

    , & author Auerbach, D

    author Barker, J. , & author Auerbach, D. ( year 1984 ). title Gas—surface interactions and dynamics; Thermal energy atomic and molecular beam studies . journal Surface Science Reports \/ , volume 4 \/ issue (1–2) , pages 1–99 . :10.1016/0167-5729(84)90005-0

    work page doi:10.1016/0167-5729(84)90005-0 1984

  6. [6]

    , author Spizzichino, A

    author Beckman, P. , author Spizzichino, A. , & author Beckmann, P. ( year 1987 ). title The scattering of electromagnetic waves from rough surfaces \/ . Radar Library. address Norwood, MA : publisher Artech House

    1987

  7. [7]

    ( year 1965 )

    author Beckmann, P. ( year 1965 ). title Shadowing of random rough surfaces . journal IEEE Transactions on Antennas and Propagation \/ , volume 13 \/ issue (3) , pages 384–388 . :10.1109/tap.1965.1138443

    work page arXiv 1965

  8. [8]

    ( year 1973 )

    author Beckmann, P. ( year 1973 ). title Scattering by non-gaussian surfaces . journal IEEE Transactions on Antennas and Propagation \/ , volume 21 \/ issue (2) , pages 169--175 . :10.1109/TAP.1973.1140444

    arXiv 1973

Show all 85 references
  1. [9]

    , & author Pilinski, M

    author Bernstein, V. , & author Pilinski, M. ( year 2022 ). title Drag coefficient constraints for space weather observations in the upper thermosphere . journal Space Weather \/ , volume 20 \/ issue (5) . :10.1029/2021sw002977

    2022 doi

  2. [10]

    title Blender - a 3D modelling and rendering package \/

    author Blender Online Community ( year 2018 ). title Blender - a 3D modelling and rendering package \/ . organization Blender Foundation address Stichting Blender Foundation, Amsterdam . http://www.blender.org

    2018

  3. [11]

    , & author Oppenheimer, R

    author Born, M. , & author Oppenheimer, R. ( year 1927 ). title Zur Quantentheorie der Molekeln . journal Annalen der Physik \/ , volume 389 \/ issue (20) , pages 457--484 . :10.1002/andp.19273892002

    1927 doi

  4. [12]

    , author Tobiska, W

    author Bowman, B. , author Tobiska, W. K. , author Marcos, F. et al. ( year 2008 ). title A New Empirical Thermospheric Density Model JB2008 Using New Solar and Geomagnetic Indices . In booktitle AIAA/AAS Astrodynamics Specialist Conference and Exhibit \/ . publisher American ...

    2008 doi

  5. [13]

    ( year 1980 )

    author Brown, G. ( year 1980 ). title Shadowing by non-Gaussian random surfaces . journal IEEE Transactions on Antennas and Propagation \/ , volume 28 \/ issue (6) , pages 788–790 . :10.1109/tap.1980.1142437

    1980

  6. [14]

    ( year 2015 )

    author Bruinsma, S. ( year 2015 ). title The dtm-2013 thermosphere model . journal Journal of Space Weather and Space Climate \/ , volume 5 \/ , pages A1 . :10.1051/swsc/2015001

    2015

  7. [15]

    , author Siemes, C

    author Bruinsma, S. , author Siemes, C. , author Emmert, J. T. et al. ( year 2023 ). title Description and comparison of 21st century thermosphere data . journal Advances in Space Research \/ , volume 72 \/ issue (12) , pages 5476–5489 . :10.1016/j.asr.2022.09.038

    2023 doi

  8. [16]

    , & author Lampis, M

    author Cercignani, C. , & author Lampis, M. ( year 1971 ). title Kinetic models for gas-surface interactions . journal Transport Theory and Statistical Physics \/ , volume 1 \/ issue (2) , pages 101–114 . :10.1080/00411457108231440

    1971 doi

  9. [17]

    , & author Michaelis, C

    author Cercignani, C. , & author Michaelis, C. ( year 2001 ). title Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations. Cambridge Texts in Applied Mathematics . journal Applied Mechanics Reviews \/ , volume 54 \/ issue (5) , pages B90--B92 . :10.1115/1.1399679

    2001 doi

  10. [18]

    , author Madden, A

    author Chen, X. , author Madden, A. S. , author Bickmore, B. R. et al. ( year 2013 ). title Dynamic weakening by nanoscale smoothing during high-velocity fault slip . journal Geology \/ , volume 41 \/ issue (7) , pages 739–742 . :10.1130/g34169.1

    2013 doi

  11. [19]

    , author Gibelli, L

    author Chen, Y. , author Gibelli, L. , & author Borg, M. K. ( year 2024 ). title Impact of random nanoscale roughness on gas-scattering dynamics . journal Phys. Rev. E \/ , volume 109 \/ , pages 065308 . :10.1103/PhysRevE.109.065308

    2024 doi

  12. [20]

    , author Gibelli, L

    author Chen, Y. , author Gibelli, L. , author Li, J. et al. ( year 2023 ). title Impact of surface physisorption on gas scattering dynamics . journal Journal of Fluid Mechanics \/ , volume 968 \/ . :10.1017/jfm.2023.496

    2023 doi

  13. [21]

    , author Fremerey, J

    author Comsa, G. , author Fremerey, J. K. , author Lindenau, B. et al. ( year 1980 ). title Calibration of a spinning rotor gas friction gauge against a fundamental vacuum pressure standard . journal Journal of Vacuum Science and Technology \/ , volume 17 \/ issue (2) , pages ...

    1980 doi

  14. [22]

    , author Cross, J

    author Cook, S. , author Cross, J. , & author Hoffbauer, M. ( year 1994 ). title AIAA--94--2637, Proceedings of the 18th Aerospace Ground Testing Conference . journal Colorado Springs \/ ,

    1994

  15. [23]

    author Cook, S. R. , & author Hoffbauer, M. A. ( year 1997 ). title Absolute momentum transfer in gas-surface scattering . journal Phys. Rev. E \/ , volume 55 \/ , pages R3828--R3831 . :10.1103/PhysRevE.55.R3828

    1997 doi

  16. [24]

    author Cook, S. R. , & author Hoffbauer, M. A. ( year 1998 ). title Analyzing gas-surface interactions using the reduced force coefficients . journal Phys. Rev. E \/ , volume 58 \/ , pages 504--511 . :10.1103/PhysRevE.58.504

    1998 doi

  17. [25]

    , author Thackeray, J

    author Cutler, C. , author Thackeray, J. W. , author Trefonas, P. et al. ( year 2021 ). title Pattern roughness analysis using power spectral density: application and impact in photoresist formulation . journal Journal of Micro/Nanopatterning, Materials, and Metrology \/ , vol...

    2021 doi

  18. [26]

    author Erofeev, A. I. ( year 1971 ). title Effect of roughness on interaction of gas flow with surface of a solid body . journal Fluid Dynamics \/ , volume 2 \/ issue (6) , pages 57–61 . :10.1007/bf01013713

    1971 doi

  19. [27]

    author Erofeev, A. I. , author Friedlander, O. G. , author Nikiforov, A. P. et al. ( year 2012 ). title The influence of roughness of the surface on the interchange of momentum between gas flow and solid surface . In booktitle AIP Conference Proceedings \/ . publisher AIP . :1...

    2012 doi

  20. [28]

    author Erofeev, A. I. , & author Nikiforov, A. P. ( year 2014 ). title Angular distribution of free molecular gas flow reflected from a solid body surface . journal TsAGI Sci. J. \/ , volume 45 \/ issue (8) , pages 927--948

    2014

  21. [29]

    , author Misture, S

    author Gong, Y. , author Misture, S. T. , author Gao, P. et al. ( year 2016 ). title Surface Roughness Measurements Using Power Spectrum Density Analysis with Enhanced Spatial Correlation Length . journal The Journal of Physical Chemistry C \/ , volume 120 \/ issue (39) , page...

    2016 doi

  22. [30]

    ( year 1965 )

    author Goodman, F. ( year 1965 ). title On the theory of accommodation coefficients—IV. Simple distribution function theory of gas-solid interaction systems . journal Journal of Physics and Chemistry of Solids \/ , volume 26 \/ issue (1) , pages 85–105 . :10.1016/0022-3697(65)90077-6

    1965 doi

  23. [31]

    author Healy, T. J. ( year 1967 ). title The scattering of particles from rough surfaces . In booktitle Fundamentals of Gas–Surface Interactions \/ (p. pages 435–447 ). publisher Elsevier . :10.1016/b978-1-4832-2901-0.50030-5

    1967 doi

  24. [32]

    , author K\" o ppen, M

    author Hellwig, M. , author K\" o ppen, M. , author Hiller, A. et al. ( year 2019 ). title Impact of Surface Roughness on Ion-Surface Interactions Studied with Energetic Carbon Ions 13C+ on Tungsten Surfaces . journal Condensed Matter \/ , volume 4 \/ issue (1) , pages 29 . :1...

    2019 doi

  25. [33]

    ( year 1939 )

    author Kac, M. ( year 1939 ). title On a Characterization of the Normal Distribution . journal American Journal of Mathematics \/ , volume 61 \/ issue (3) , pages 726 . :10.2307/2371328

    1939 doi

  26. [34]

    , & author Mallik, B

    author Karan, S. , & author Mallik, B. ( year 2008 ). title Power spectral density analysis and photoconducting behavior in copper(ii) phthalocyanine nanostructured thin films . journal Physical Chemistry Chemical Physics \/ , volume 10 \/ issue (45) , pages 6751 . :10.1039/b809648a

    2008 doi

  27. [35]

    author Kleyn, A. W. ( year 2003 ). title Molecular beams and chemical dynamics at surfaces . journal Chemical Society Reviews \/ , volume 32 \/ issue (2) , pages 87–95 . :10.1039/b105760j

    2003 doi

  28. [36]

    author Knechtel, E. D. , & author Pitts, W. C. ( year 1973 ). title Normal and tangential momentum accommodation for earth satellite conditions . journal Astronautica Acta \/ , volume 18 \/ issue (NAS 1.15: 112784)

    1973

  29. [37]

    ( year 1999 )

    author Koch, K.-R. ( year 1999 ). title Parameter Estimation and Hypothesis Testing in Linear Models \/ . address Berlin : publisher Springer . :10.1007/978-3-662-03976-2

    1999 doi

  30. [38]

    , author Li, Q

    author Liang, T. , author Li, Q. , & author Ye, W. ( year 2018 ). title A physical-based gas–surface interaction model for rarefied gas flow simulation . journal Journal of Computational Physics \/ , volume 352 \/ , pages 105–122 . :10.1016/j.jcp.2017.08.061

    2018 doi

  31. [39]

    , author Zhang, J

    author Liang, T. , author Zhang, J. , & author Li, Q. ( year 2021 ). title A parameter-free physical model for gas–surface interaction . journal Physics of Fluids \/ , volume 33 \/ issue (8) . :10.1063/5.0059029

    2021 doi

  32. [40]

    , author Grenier, R

    author Liao, M. , author Grenier, R. , author To, Q.-D. et al. ( year 2018 ). title Helium and Argon Interactions with Gold Surfaces: Ab Initio-Assisted Determination of the He–Au Pairwise Potential and Its Application to Accommodation Coefficient Determination . journal The J...

    2018 doi

  33. [41]

    author Litvak, M. Y. , & author Malyugin, V. I. ( year 2012 ). title Poly-Gaussian models of a non-Gaussian randomly rough surface . journal Technical Physics \/ , volume 57 \/ issue (4) , pages 524–533 . :10.1134/s1063784212040172

    2012 doi

  34. [42]

    , author Sharma, F

    author Liu, S.-M. , author Sharma, F. K. , & author Knuth, E. L. ( year 1979 ). title Satellite drag coefficients calculated from measured distributions of reflected helium atoms . journal AIAA Journal \/ , volume 17 \/ issue (12) , pages 1314–1319 . :10.2514/3.7629

    1979 doi

  35. [43]

    , author Crisp, N

    author Livadiotti, S. , author Crisp, N. H. , author Roberts, P. C. et al. ( year 2020 ). title A review of gas-surface interaction models for orbital aerodynamics applications . journal Progress in Aerospace Sciences \/ , volume 119 \/ , pages 100675 . :10.1016/j.paerosci.2020.100675

    2020

  36. [44]

    author Logan, R. M. , & author Keck, J. C. ( year 1968 ). title Classical Theory for the Interaction of Gas Atoms with Solid Surfaces . journal The Journal of Chemical Physics \/ , volume 49 \/ issue (2) , pages 860–876 . :10.1063/1.1670153

    1968 doi

  37. [45]

    author Logan, R. M. , & author Stickney, R. E. ( year 1966 ). title Simple Classical Model for the Scattering of Gas Atoms from a Solid Surface . journal The Journal of Chemical Physics \/ , volume 44 \/ issue (1) , pages 195–201 . :10.1063/1.1726446

    1966 doi

  38. [46]

    author Lord, R. G. ( year 1995 ). title Some further extensions of the Cercignani–Lampis gas–surface interaction model . journal Physics of Fluids \/ , volume 7 \/ issue (5) , pages 1159–1161 . :10.1063/1.868557

    1995 doi

  39. [47]

    author MacKay, D. J. C. ( year 2003 ). title Information theory, inference and learning algorithms \/ . address Cambridge, England : publisher Cambridge University Press

    2003

  40. [48]

    , author Doornbos, E

    author March, G. , author Doornbos, E. , & author Visser, P. ( year 2019 ). title High-fidelity geometry models for improving the consistency of CHAMP, GRACE, GOCE and Swarm thermospheric density data sets . journal Advances in Space Research \/ , volume 63 \/ issue (1) , page...

    2019 doi

  41. [49]

    , author van den IJssel, J

    author March, G. , author van den IJssel, J. , author Siemes, C. et al. ( year 2021 ). title Gas-surface interactions modelling influence on satellite aerodynamics and thermosphere mass density . journal Journal of Space Weather and Space Climate \/ , volume 11 \/ , pages 54 ....

    2021

  42. [50]

    , author Kerwin, J

    author Mateljevic, N. , author Kerwin, J. , author Roy, S. et al. ( year 2009 ). title Accommodation of gases at rough surfaces . journal The Journal of Physical Chemistry C \/ , volume 113 \/ issue (6) , pages 2360–2367 . :10.1021/jp8077634

    2009 doi

  43. [51]

    author Maxwell, J. C. ( year 1879 ). title On stresses in rarified gases arising from inequalities of temperature . journal Philosophical Transactions of the Royal Society of London \/ , volume 170 \/ , pages 231–256 . :10.1098/rstl.1879.0067

  44. [52]

    author McLaughlin, C. A. , author Mance, S. , & author Lichtenberg, T. ( year 2011 ). title Drag Coefficient Estimation in Orbit Determination . journal The Journal of the Astronautical Sciences \/ , volume 58 \/ issue (3) , pages 513–530 . :10.1007/bf03321183

    2011 doi

  45. [53]

    author Mehta, P. M. , & author Linares, R. ( year 2018 ). title A New Transformative Framework for Data Assimilation and Calibration of Physical Ionosphere‐Thermosphere Models . journal Space Weather \/ , volume 16 \/ issue (8) , pages 1086–1100 . :10.1029/2018sw001875

    2018 doi

  46. [54]

    author Mehta, P. M. , author Paul, S. N. , author Crisp, N. H. et al. ( year 2023 ). title Satellite drag coefficient modeling for thermosphere science and mission operations . journal Advances in Space Research \/ , volume 72 \/ issue (12) , pages 5443–5459 . :10.1016/j.asr.2...

    2023 doi

  47. [55]

    author Mehta, P. M. , author Walker, A. , author McLaughlin, C. A. et al. ( year 2014 ). title Comparing physical drag coefficients computed using different gas–surface interaction models . journal Journal of Spacecraft and Rockets \/ , volume 51 \/ issue (3) , pages 873–883 ....

    2014 doi

  48. [56]

    author Mehta, P. M. , author Walker, A. C. , author Sutton, E. K. et al. ( year 2017 ). title New density estimates derived using accelerometers on board the CHAMP and GRACE satellites . journal Space Weather \/ , volume 15 \/ issue (4) , pages 558–576 . :10.1002/2016sw001562

    2017 doi

  49. [57]

    , author Rosenbluth, A

    author Metropolis, N. , author Rosenbluth, A. W. , author Rosenbluth, M. N. et al. ( year 1953 ). title Equation of State Calculations by Fast Computing Machines . journal The Journal of Chemical Physics \/ , volume 21 \/ issue (6) , pages 1087–1092 . :10.1063/1.1699114

    1953 doi

  50. [58]

    , & author Moe, M

    author Moe, K. , & author Moe, M. M. ( year 2005 ). title Gas–surface interactions and satellite drag coefficients . journal Planetary and Space Science \/ , volume 53 \/ issue (8) , pages 793–801 . :10.1016/j.pss.2005.03.005

    2005 doi

  51. [59]

    , author Moe, M

    author Moe, K. , author Moe, M. M. , & author Yelaca, N. W. ( year 1972 ). title Effect of surface heterogeneity on the adsorptive behavior of orbiting pressure gages . journal Journal of Geophysical Research \/ , volume 77 \/ issue (22) , pages 4242–4247 . :10.1029/ja077i022p04242

    1972 doi

  52. [60]

    author Moe, M. M. , author Wallace, S. D. , & author Moe, K. ( year 1993 ). title Refinements in determining satellite drag coefficients - Method for resolving density discrepancies . journal Journal of Guidance, Control, and Dynamics \/ , volume 16 \/ issue (3) , pages 441–44...

    1993 doi

  53. [61]

    author Murray, V. J. , author Marshall, B. C. , author Woodburn, P. J. et al. ( year 2015 ). title Inelastic and Reactive Scattering Dynamics of Hyperthermal O and O2 on Hot Vitreous Carbon Surfaces . journal The Journal of Physical Chemistry C \/ , volume 119 \/ issue (26) , ...

    2015 doi

  54. [62]

    author Murray, V. J. , author Pilinski, M. D. , author Smoll, E. J. et al. ( year 2017 ). title Gas–Surface Scattering Dynamics Applied to Concentration of Gases for Mass Spectrometry in Tenuous Atmospheres . journal The Journal of Physical Chemistry C \/ , volume 121 \/ issue...

    2017 doi

  55. [63]

    author Mwema, F. M. , author Oladijo, O. P. , author Sathiaraj, T. S. et al. ( year 2018 ). title Atomic force microscopy analysis of surface topography of pure thin aluminum films . journal Materials Research Express \/ , volume 5 \/ issue (4) , pages 046416 . :10.1088/2053-1...

    2018 doi

  56. [64]

    , author Leu, T

    author Ozhgibesov, M. , author Leu, T. , author Cheng, C. et al. ( year 2013 ). title Studies on argon collisions with smooth and rough tungsten surfaces . journal Journal of Molecular Graphics and Modelling \/ , volume 45 \/ , pages 45–49 . :10.1016/j.jmgm.2013.08.010

    2013 doi

  57. [65]

    , & author Boyd, I

    author Padilla, J. , & author Boyd, I. ( year 2007 ). title Assessment of Gas-Surface Interaction Models in DSMC Analysis of Rarefied Hypersonic Flow . In booktitle 39th AIAA Thermophysics Conference \/ . publisher American Institute of Aeronautics and Astronautics . :10.2514/...

    2007 doi

  58. [66]

    , author Anselmo, L

    author Pardini, C. , author Anselmo, L. , author Moe, K. et al. ( year 2010 ). title Drag and energy accommodation coefficients during sunspot maximum . journal Advances in Space Research \/ , volume 45 \/ issue (5) , pages 638–650 . :10.1016/j.asr.2009.08.034

    2010 doi

  59. [67]

    , author Tobiska, W

    author Pardini, C. , author Tobiska, W. K. , & author Anselmo, L. ( year 2006 ). title Analysis of the orbital decay of spherical satellites using different solar flux proxies and atmospheric density models . journal Advances in Space Research \/ , volume 37 \/ issue (2) , pag...

    2006 doi

  60. [68]

    author Picone, J. M. , author Hedin, A. E. , author Drob, D. P. et al. ( year 2002 ). title Nrlmsise‐00 empirical model of the atmosphere: Statistical comparisons and scientific issues . journal Journal of Geophysical Research: Space Physics \/ , volume 107 \/ issue (A12) . :1...

    2002 doi

  61. [69]

    author Pilinski, M. D. , author Argrow, B. M. , author Palo, S. E. et al. ( year 2013 ). title Semi-Empirical Satellite Accommodation Model for Spherical and Randomly Tumbling Objects . journal Journal of Spacecraft and Rockets \/ , volume 50 \/ issue (3) , pages 556–571 . :10...

    2013 doi

  62. [70]

    , & author Tatchen, J

    author Pollak, E. , & author Tatchen, J. ( year 2009 ). title Rainbow scattering of argon from 2H -W (100) . journal Phys. Rev. B \/ , volume 80 \/ , pages 115404 . :10.1103/PhysRevB.80.115404

    2009 doi

  63. [71]

    author Rettner, C. T. , author Barker, J. A. , & author Bethune, D. S. ( year 1991 ). title Angular and velocity distributions characteristic of the transition between the thermal and structure regimes of gas-surface scattering . journal Physical Review Letters \/ , volume 67 ...

    1991 doi

  64. [72]

    author Roman, M. J. , author Knight, A. G. , author Moon, D. R. et al. ( year 2023 ). title Inelastic scattering of OH from a liquid PFPE surface: Resolution of correlated speed and angular distributions . journal The Journal of Chemical Physics \/ , volume 158 \/ issue (24) ....

    2023 doi

  65. [73]

    author Sazhin, O. V. , author Borisov, S. F. , & author Sharipov, F. ( year 2001 ). title Accommodation coefficient of tangential momentum on atomically clean and contaminated surfaces . journal Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films \/ , volume ...

    2001 doi

  66. [74]

    author Sentman, L. H. ( year 1961 ). title FREE MOLECULE FLOW THEORY AND ITS APPLICATION TO THE DETERMINATION OF AERODYNAMIC FORCES . https://api.semanticscholar.org/CorpusID:92321666

    1961

  67. [75]

    , author Kano, N

    author Shoda, K. , author Kano, N. , author Jotaki, Y. et al. ( year 2022 ). title Anisotropic molecular scattering at microstructured surface for rarefied gas compression inside air breathing ion engine . journal CEAS Space Journal \/ , volume 15 \/ issue (3) , pages 403–411 ...

    2022 doi

  68. [76]

    , author Wu, X

    author Shu, C. , author Wu, X. , author Zhong, M. et al. ( year 2023 ). title The atomic oxygen resistant study of a transparent polyimide film containing phosphorus and fluorine . journal Applied Surface Science \/ , volume 631 \/ , pages 157562 . :10.1016/j.apsusc.2023.157562

    2023

  69. [77]

    , author Borries, C

    author Siemes, C. , author Borries, C. , author Bruinsma, S. et al. ( year 2023 ). title New thermosphere neutral mass density and crosswind datasets from champ, grace, and grace-fo . journal Journal of Space Weather and Space Climate \/ , volume 13 \/ , pages 16 . :10.1051/sw...

    2023

  70. [78]

    , author den IJssel , J

    author Siemes, C. , author den IJssel , J. , & author Visser, P. ( year 2024 ). title Uncertainty of thermosphere mass density observations derived from accelerometer and gnss tracking data . journal Advances in Space Research \/ , . :https://doi.org/10.1016/j.asr.2024.02.057

    2024 doi

  71. [79]

    ( year 1967 )

    author Smith, B. ( year 1967 ). title Geometrical shadowing of a random rough surface . journal IEEE Transactions on Antennas and Propagation \/ , volume 15 \/ issue (5) , pages 668–671 . :10.1109/tap.1967.1138991

    1967

  72. [80]

    author Tully, J. C. ( year 1990 ). title Washboard model of gas–surface scattering . journal The Journal of Chemical Physics \/ , volume 92 \/ issue (1) , pages 680–686 . :10.1063/1.458421

    1990 doi

  73. [81]

    author Vallado, D. A. ( year 2007 ). title Fundamentals of Astrodynamics and Applications \/ . ( edition 2nd ed.). publisher Springer-Netherlands

    2007

  74. [82]

    , author Mehta, P

    author Walker, A. , author Mehta, P. , & author Koller, J. ( year 2014 a ). title Drag Coefficient Model Using the Cercignani–Lampis–Lord Gas–Surface Interaction Model . journal Journal of Spacecraft and Rockets \/ , volume 51 \/ issue (5) , pages 1544–1563 . :10.2514/1.a32677

    2014 doi

  75. [83]

    , author Mehta, P

    author Walker, A. , author Mehta, P. , & author Koller, J. ( year 2014 b ). title The effect of different adsorption models on satellite drag coefficients . In booktitle Astrodynamics 2013 - Advances in the Astronautical Sciences \/ Advances in the Astronautical Sciences (pp. ...

    2014

  76. [84]

    ( year 1930 )

    author Wiener, N. ( year 1930 ). title Generalized harmonic analysis . journal Acta Mathematica \/ , volume 55 \/ issue (0) , pages 117–258 . :10.1007/bf02546511

    1930 doi

  77. [85]

    , author Murray, V

    author Xu, C. , author Murray, V. J. , author Pilinski, M. D. et al. ( year 2023 ). title Gas concentration in rarefied flows: Experiments and modeling . journal Aerospace Science and Technology \/ , volume 142 \/ , pages 108568 . :10.1016/j.ast.2023.108568

    2023

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.