A Differential Rotation Law for Stars and Fluid Planets

Joel Uriel Cisneros-Parra; Francisco Javier Martínez-Herrera; Daniel Montalvo-Castro

doi:doi:10.11648/j.ajaa.20200802.13

| Peer-Reviewed

A Differential Rotation Law for Stars and Fluid Planets

Joel Uriel Cisneros-Parra, Francisco Javier Martínez-Herrera, Daniel Montalvo-Castro

Published in American Journal of Astronomy and Astrophysics (Volume 8, Issue 2)

Received: 31 March 2020 Accepted: 3 May 2020 Published: 15 May 2020

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

We derive, supported on a generalization of Bernoulli’s equation, a law of rotation for any axial-symmetric, self-gravitating fluid mass. For a homogeneous mass, the law depends solely on the derivative of the potential with respect to the distance to the rotation axis, implying generally differential rotation, the Maclaurin spheroids representing the only case of solid-body rotation. We turn then to a heterogeneous mass consisting of any number l of concentric layers, each of constant density, finding that the angular velocity profile of a given layer depends on that of the layer immediately above it. Finally, we let l tend to infinity to convert our model into continuous mass distribution, the result being a certain rotation profile for the surface, and law of differential rotation change at its interior. To support the fundamentals of our approach, we write the potential integrals for the three mass distributions. The aim of a continuous distribution is that it may facilitate a comparison---to be carried out in a future paper---between our results and those of other researchers who employ structure equations. We point out that the distribution of angular velocity is a consequence of the equilibrium, rather than being imposed ad initio. The law was used in a past paper to construct a Jupiter multi-layer model adopting the spheroidal (a distorted spheroid) shape for each of the layers, taking as reference the gravitational data surveyed by the Juno mission. The procedure used here is not restricted to axial-symmetric cases.

Published in	American Journal of Astronomy and Astrophysics (Volume 8, Issue 2)
DOI	10.11648/j.ajaa.20200802.13
Page(s)	30-34
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Gravitation, Hydrodynamics, Planets and Satellites, Stars, Rotation

References

[1]	Cisneros, J. U., F. J. Martínez, & D. Montalvo, RMx, AA, 2016, 52, 375.
[2]	Cisneros, J. U., F. J. Martínez, & D. Montalvo, 2017, ApJ, 848, 109.
[3]	Kitchatinov, L. L., 2005, Physics-Uspekhi, 48 (5), 449.
[4]	Cisneros, J. U., F. J. Martínez, & D. Montalvo, American Journal of Astronomy and Astrophysics. Vol. 8, No. 1, 2020, pp. 8-14.
[5]	Jeans, J. H. 1919, Phil Trans. R. Soc., (Cambridge, England Cambridge University Press).
[6]	Poincare, H., Theorie des Turbillons, (1893), Gauthier-Villars, Paris, 212.
[7]	Landau, L.\ D.\ \&Lifshitz E.\ M., 1987, Course of Theoretical Physics: Fluid Mechanics, New York: Pergamon Press.
[8]	Tassoul, J. L., Stellar Rotation, (2000), Cambridge University Press, 256.
[9]	Hubbard, W. B., 2012, ApJL, 756: L15.
[10]	Jacobi, C. G. J., Uber die Figur des Gleichgewichts, Poggendorff Annallen der Physik und Chemie, Vol. 33, 1834, pp. 229-238; reprinted in Gesammelte Werke, Vol. 2, pp. 17-72, G. Reimer, Berlin, 1882.
[11]	Lyttleton, R. A. 1953, The Stability of Rotating LiquidMasses (Cambridge: Cambridge University Press).
[12]	Chandrasekhar, S. 1969 Ellipsoidal Figures Of Equilibrium (Yale: University Press).
[13]	Dedekind, R., Zusatz zu der vorstehenden Abhandlung, J. Reine Angew. Math., Vol. 58, 1860.
[14]	Ostriker, J. P., J. W-K. Mark, 1968, ApJ, 151.
[15]	Tassoul, J. L., J. P. Ostriker ApJ, 1968, 154, 613.

Cite This Article

Plain Text BibTeX RIS

APA Style

Joel Uriel Cisneros-Parra, Francisco Javier Martínez-Herrera, Daniel Montalvo-Castro. (2020). A Differential Rotation Law for Stars and Fluid Planets. American Journal of Astronomy and Astrophysics, 8(2), 30-34. https://doi.org/10.11648/j.ajaa.20200802.13

Copy | Download

ACS Style

Joel Uriel Cisneros-Parra; Francisco Javier Martínez-Herrera; Daniel Montalvo-Castro. A Differential Rotation Law for Stars and Fluid Planets. Am. J. Astron. Astrophys. 2020, 8(2), 30-34. doi: 10.11648/j.ajaa.20200802.13

Copy | Download

AMA Style

Joel Uriel Cisneros-Parra, Francisco Javier Martínez-Herrera, Daniel Montalvo-Castro. A Differential Rotation Law for Stars and Fluid Planets. Am J Astron Astrophys. 2020;8(2):30-34. doi: 10.11648/j.ajaa.20200802.13

Copy | Download

@article{10.11648/j.ajaa.20200802.13,
  author = {Joel Uriel Cisneros-Parra and Francisco Javier Martínez-Herrera and Daniel Montalvo-Castro},
  title = {A Differential Rotation Law for Stars and Fluid Planets},
  journal = {American Journal of Astronomy and Astrophysics},
  volume = {8},
  number = {2},
  pages = {30-34},
  doi = {10.11648/j.ajaa.20200802.13},
  url = {https://doi.org/10.11648/j.ajaa.20200802.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajaa.20200802.13},
  abstract = {We derive, supported on a generalization of Bernoulli’s equation, a law of rotation for any axial-symmetric, self-gravitating fluid mass. For a homogeneous mass, the law depends solely on the derivative of the potential with respect to the distance to the rotation axis, implying generally differential rotation, the Maclaurin spheroids representing the only case of solid-body rotation. We turn then to a heterogeneous mass consisting of any number l of concentric layers, each of constant density, finding that the angular velocity profile of a given layer depends on that of the layer immediately above it. Finally, we let l tend to infinity to convert our model into continuous mass distribution, the result being a certain rotation profile for the surface, and law of differential rotation change at its interior. To support the fundamentals of our approach, we write the potential integrals for the three mass distributions. The aim of a continuous distribution is that it may facilitate a comparison---to be carried out in a future paper---between our results and those of other researchers who employ structure equations. We point out that the distribution of angular velocity is a consequence of the equilibrium, rather than being imposed ad initio. The law was used in a past paper to construct a Jupiter multi-layer model adopting the spheroidal (a distorted spheroid) shape for each of the layers, taking as reference the gravitational data surveyed by the Juno mission. The procedure used here is not restricted to axial-symmetric cases.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - A Differential Rotation Law for Stars and Fluid Planets
AU  - Joel Uriel Cisneros-Parra
AU  - Francisco Javier Martínez-Herrera
AU  - Daniel Montalvo-Castro
Y1  - 2020/05/15
PY  - 2020
N1  - https://doi.org/10.11648/j.ajaa.20200802.13
DO  - 10.11648/j.ajaa.20200802.13
T2  - American Journal of Astronomy and Astrophysics
JF  - American Journal of Astronomy and Astrophysics
JO  - American Journal of Astronomy and Astrophysics
SP  - 30
EP  - 34
PB  - Science Publishing Group
SN  - 2376-4686
UR  - https://doi.org/10.11648/j.ajaa.20200802.13
AB  - We derive, supported on a generalization of Bernoulli’s equation, a law of rotation for any axial-symmetric, self-gravitating fluid mass. For a homogeneous mass, the law depends solely on the derivative of the potential with respect to the distance to the rotation axis, implying generally differential rotation, the Maclaurin spheroids representing the only case of solid-body rotation. We turn then to a heterogeneous mass consisting of any number l of concentric layers, each of constant density, finding that the angular velocity profile of a given layer depends on that of the layer immediately above it. Finally, we let l tend to infinity to convert our model into continuous mass distribution, the result being a certain rotation profile for the surface, and law of differential rotation change at its interior. To support the fundamentals of our approach, we write the potential integrals for the three mass distributions. The aim of a continuous distribution is that it may facilitate a comparison---to be carried out in a future paper---between our results and those of other researchers who employ structure equations. We point out that the distribution of angular velocity is a consequence of the equilibrium, rather than being imposed ad initio. The law was used in a past paper to construct a Jupiter multi-layer model adopting the spheroidal (a distorted spheroid) shape for each of the layers, taking as reference the gravitational data surveyed by the Juno mission. The procedure used here is not restricted to axial-symmetric cases.
VL  - 8
IS  - 2
ER  -

Copy | Download

Author Information

Joel Uriel Cisneros-Parra

Science Faculty, Autonomous University of San Luis Potosí, San Luis Potosí, México
Francisco Javier Martínez-Herrera

Physics Institute, Autonomous University of San Luis Potosí, San Luis Potosí, México
Daniel Montalvo-Castro

Physics Institute, Autonomous University of San Luis Potosí, San Luis Potosí, México

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Joel Uriel Cisneros-Parra, Francisco Javier Martínez-Herrera, Daniel Montalvo-Castro. (2020). A Differential Rotation Law for Stars and Fluid Planets. American Journal of Astronomy and Astrophysics, 8(2), 30-34. https://doi.org/10.11648/j.ajaa.20200802.13

Copy | Download

ACS Style

Joel Uriel Cisneros-Parra; Francisco Javier Martínez-Herrera; Daniel Montalvo-Castro. A Differential Rotation Law for Stars and Fluid Planets. Am. J. Astron. Astrophys. 2020, 8(2), 30-34. doi: 10.11648/j.ajaa.20200802.13

Copy | Download

AMA Style

Joel Uriel Cisneros-Parra, Francisco Javier Martínez-Herrera, Daniel Montalvo-Castro. A Differential Rotation Law for Stars and Fluid Planets. Am J Astron Astrophys. 2020;8(2):30-34. doi: 10.11648/j.ajaa.20200802.13

Copy | Download

@article{10.11648/j.ajaa.20200802.13,
  author = {Joel Uriel Cisneros-Parra and Francisco Javier Martínez-Herrera and Daniel Montalvo-Castro},
  title = {A Differential Rotation Law for Stars and Fluid Planets},
  journal = {American Journal of Astronomy and Astrophysics},
  volume = {8},
  number = {2},
  pages = {30-34},
  doi = {10.11648/j.ajaa.20200802.13},
  url = {https://doi.org/10.11648/j.ajaa.20200802.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajaa.20200802.13},
  abstract = {We derive, supported on a generalization of Bernoulli’s equation, a law of rotation for any axial-symmetric, self-gravitating fluid mass. For a homogeneous mass, the law depends solely on the derivative of the potential with respect to the distance to the rotation axis, implying generally differential rotation, the Maclaurin spheroids representing the only case of solid-body rotation. We turn then to a heterogeneous mass consisting of any number l of concentric layers, each of constant density, finding that the angular velocity profile of a given layer depends on that of the layer immediately above it. Finally, we let l tend to infinity to convert our model into continuous mass distribution, the result being a certain rotation profile for the surface, and law of differential rotation change at its interior. To support the fundamentals of our approach, we write the potential integrals for the three mass distributions. The aim of a continuous distribution is that it may facilitate a comparison---to be carried out in a future paper---between our results and those of other researchers who employ structure equations. We point out that the distribution of angular velocity is a consequence of the equilibrium, rather than being imposed ad initio. The law was used in a past paper to construct a Jupiter multi-layer model adopting the spheroidal (a distorted spheroid) shape for each of the layers, taking as reference the gravitational data surveyed by the Juno mission. The procedure used here is not restricted to axial-symmetric cases.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - A Differential Rotation Law for Stars and Fluid Planets
AU  - Joel Uriel Cisneros-Parra
AU  - Francisco Javier Martínez-Herrera
AU  - Daniel Montalvo-Castro
Y1  - 2020/05/15
PY  - 2020
N1  - https://doi.org/10.11648/j.ajaa.20200802.13
DO  - 10.11648/j.ajaa.20200802.13
T2  - American Journal of Astronomy and Astrophysics
JF  - American Journal of Astronomy and Astrophysics
JO  - American Journal of Astronomy and Astrophysics
SP  - 30
EP  - 34
PB  - Science Publishing Group
SN  - 2376-4686
UR  - https://doi.org/10.11648/j.ajaa.20200802.13
AB  - We derive, supported on a generalization of Bernoulli’s equation, a law of rotation for any axial-symmetric, self-gravitating fluid mass. For a homogeneous mass, the law depends solely on the derivative of the potential with respect to the distance to the rotation axis, implying generally differential rotation, the Maclaurin spheroids representing the only case of solid-body rotation. We turn then to a heterogeneous mass consisting of any number l of concentric layers, each of constant density, finding that the angular velocity profile of a given layer depends on that of the layer immediately above it. Finally, we let l tend to infinity to convert our model into continuous mass distribution, the result being a certain rotation profile for the surface, and law of differential rotation change at its interior. To support the fundamentals of our approach, we write the potential integrals for the three mass distributions. The aim of a continuous distribution is that it may facilitate a comparison---to be carried out in a future paper---between our results and those of other researchers who employ structure equations. We point out that the distribution of angular velocity is a consequence of the equilibrium, rather than being imposed ad initio. The law was used in a past paper to construct a Jupiter multi-layer model adopting the spheroidal (a distorted spheroid) shape for each of the layers, taking as reference the gravitational data surveyed by the Juno mission. The procedure used here is not restricted to axial-symmetric cases.
VL  - 8
IS  - 2
ER  -

Copy | Download