An iterative method for obtaining a nonlinear solution for the temperature distribution of a rotating spherical body revolving in a circular orbit around a star

M. Sekiya, A. A. Shimoda

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2 Citations (Scopus)

Abstract

We developed an iterative method for determining the time-dependent three-dimensional temperature distribution in a spherical body with smooth surface that is irradiated by a star. In the method developed in our previous paper (Sekiya et al.; 2012), only the rotational motion is taken into account and the effect due to the revolution around the star is ignored. The present work includes both the effects of the rotation and the revolution. We take into account the cooling due to the surface radiation that is proportional to the fourth power of the temperature; this is the difference in the present work from Vokrouhlický (1999) that employs the linear approximation for the radiative cooling. It is assumed that material parameters such as the thermal conductivity and the thermometric conductivity are constant throughout the spherical body. We obtain a general solution for the temperature distribution inside a body by using the spherical harmonics and the spherical Bessel functions for space and the Fourier series for the time. The term in the boundary condition that represents the heating due to the star is also expanded into the spherical harmonics and the Fourier series. The coefficients of the general solution are fitted to satisfy the surface boundary condition by using an iterative method. We obtained solutions that satisfy the nonlinear boundary condition within 0.1% accuracy. The temperature distribution determined according to the iterative method is different from that according to the linear approximation; both the maximum and minimum temperatures at a given time after the summer solstice for an iterative solution are lower than those for a linear solution. The maximum difference between rate of change of the semimajor axis due to the Yarkovsky effect according to the iterative solution and that according to the linear solution is about 20%. Therefore, current understanding of the Yarkovsky effect based on linear solutions is fairly good.

Original languageEnglish
Pages (from-to)112-121
Number of pages10
JournalPlanetary and Space Science
Volume84
DOIs
Publication statusPublished - Aug 1 2013

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circular orbits
temperature distribution
stars
iterative solution
Fourier series
spherical harmonics
boundary conditions
boundary condition
temperature
cooling
solstices
Bessel functions
approximation
summer
thermal conductivity
conductivity
heating
distribution
method
radiation

All Science Journal Classification (ASJC) codes

  • Astronomy and Astrophysics
  • Space and Planetary Science

Cite this

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title = "An iterative method for obtaining a nonlinear solution for the temperature distribution of a rotating spherical body revolving in a circular orbit around a star",
abstract = "We developed an iterative method for determining the time-dependent three-dimensional temperature distribution in a spherical body with smooth surface that is irradiated by a star. In the method developed in our previous paper (Sekiya et al.; 2012), only the rotational motion is taken into account and the effect due to the revolution around the star is ignored. The present work includes both the effects of the rotation and the revolution. We take into account the cooling due to the surface radiation that is proportional to the fourth power of the temperature; this is the difference in the present work from Vokrouhlick{\'y} (1999) that employs the linear approximation for the radiative cooling. It is assumed that material parameters such as the thermal conductivity and the thermometric conductivity are constant throughout the spherical body. We obtain a general solution for the temperature distribution inside a body by using the spherical harmonics and the spherical Bessel functions for space and the Fourier series for the time. The term in the boundary condition that represents the heating due to the star is also expanded into the spherical harmonics and the Fourier series. The coefficients of the general solution are fitted to satisfy the surface boundary condition by using an iterative method. We obtained solutions that satisfy the nonlinear boundary condition within 0.1{\%} accuracy. The temperature distribution determined according to the iterative method is different from that according to the linear approximation; both the maximum and minimum temperatures at a given time after the summer solstice for an iterative solution are lower than those for a linear solution. The maximum difference between rate of change of the semimajor axis due to the Yarkovsky effect according to the iterative solution and that according to the linear solution is about 20{\%}. Therefore, current understanding of the Yarkovsky effect based on linear solutions is fairly good.",
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N2 - We developed an iterative method for determining the time-dependent three-dimensional temperature distribution in a spherical body with smooth surface that is irradiated by a star. In the method developed in our previous paper (Sekiya et al.; 2012), only the rotational motion is taken into account and the effect due to the revolution around the star is ignored. The present work includes both the effects of the rotation and the revolution. We take into account the cooling due to the surface radiation that is proportional to the fourth power of the temperature; this is the difference in the present work from Vokrouhlický (1999) that employs the linear approximation for the radiative cooling. It is assumed that material parameters such as the thermal conductivity and the thermometric conductivity are constant throughout the spherical body. We obtain a general solution for the temperature distribution inside a body by using the spherical harmonics and the spherical Bessel functions for space and the Fourier series for the time. The term in the boundary condition that represents the heating due to the star is also expanded into the spherical harmonics and the Fourier series. The coefficients of the general solution are fitted to satisfy the surface boundary condition by using an iterative method. We obtained solutions that satisfy the nonlinear boundary condition within 0.1% accuracy. The temperature distribution determined according to the iterative method is different from that according to the linear approximation; both the maximum and minimum temperatures at a given time after the summer solstice for an iterative solution are lower than those for a linear solution. The maximum difference between rate of change of the semimajor axis due to the Yarkovsky effect according to the iterative solution and that according to the linear solution is about 20%. Therefore, current understanding of the Yarkovsky effect based on linear solutions is fairly good.

AB - We developed an iterative method for determining the time-dependent three-dimensional temperature distribution in a spherical body with smooth surface that is irradiated by a star. In the method developed in our previous paper (Sekiya et al.; 2012), only the rotational motion is taken into account and the effect due to the revolution around the star is ignored. The present work includes both the effects of the rotation and the revolution. We take into account the cooling due to the surface radiation that is proportional to the fourth power of the temperature; this is the difference in the present work from Vokrouhlický (1999) that employs the linear approximation for the radiative cooling. It is assumed that material parameters such as the thermal conductivity and the thermometric conductivity are constant throughout the spherical body. We obtain a general solution for the temperature distribution inside a body by using the spherical harmonics and the spherical Bessel functions for space and the Fourier series for the time. The term in the boundary condition that represents the heating due to the star is also expanded into the spherical harmonics and the Fourier series. The coefficients of the general solution are fitted to satisfy the surface boundary condition by using an iterative method. We obtained solutions that satisfy the nonlinear boundary condition within 0.1% accuracy. The temperature distribution determined according to the iterative method is different from that according to the linear approximation; both the maximum and minimum temperatures at a given time after the summer solstice for an iterative solution are lower than those for a linear solution. The maximum difference between rate of change of the semimajor axis due to the Yarkovsky effect according to the iterative solution and that according to the linear solution is about 20%. Therefore, current understanding of the Yarkovsky effect based on linear solutions is fairly good.

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