The purpose of this work is to study the phenomenon of tidal locking in a pedagogical framework by analyzing the effective gravitational potential of a two-body system with two spinning objects. It is shown that the effective potential of such a system is an example of a fold catastrophe. In fact, the existence of a local minimum and saddle point, corresponding to tidally locked circular orbits, is regulated by a single dimensionless control parameter that depends on the properties of the two bodies and on the total angular momentum of the system. The method described in this work results in compact expressions for the radius of the circular orbit and the tidally locked spin/orbital frequency. The limiting case in which one of the two orbiting objects is point-like is studied in detail. An analysis of the effective potential, which in this limit depends on only two parameters, allows one to clearly visualize the properties of the system. The notorious case of the Mars' moon Phobos is presented as an example of a satellite that is past the no return point and, therefore, will not reach a stable or unstable tidally locked orbit.
REFERENCES
1.
Encyclopedia of Astronomy and
Astrophysics
, edited by
Paul
Murdin
( Institute of Physics
Publishing
, Bristol
, 2001
),
p. 3998
.2.
3.
Sir Isaac
Newton
, Philosophiæ Naturalis Principia
Mathematica
(1687
).4.
5.
A. B.
Arons
, “ Basic physics of the semidiurnal lunar
tide
,” Am. J. Phys.
47
, 934
–937
(1979
).6.
O.
Grøn
, “ A tidal force pendulum
,” Am. J.
Phys.
51
, 429
–431
(1983
).7.
Gary
White
,
Tony
Mondragon
,
David
Slaughter
, and
Dorothy
Coates
, “ Modelling tidal effects
,” Am. J.
Phys.
61
, 367
–371
(1993
).8.
Mitchell M.
Withers
, “ Why do tides exist?
,” Phys.
Teach.
31
, 394
–398
(1993
).9.
M. A.
Koenders
, “ The effects of tidal forces on an elastic satellite in a
closed orbit
,” Eur. J. Phys.
19
, 265
–270
(1998
).10.
Eugene I.
Butikov
, “ A dynamical picture of the oceanic
tides
,” Am. J. Phys.
70
, 1001
–1011
(2002
).11.
H.
Razmi
, “ On the tidal force of the Moon on the
Earth
,” Eur. J. Phys.
26
, 927
–934
(2005
).12.
Marco
Masi
, “ On compressive radial tidal forces
,”
Am. J. Phys.
75
, 116
–124
(2007
).13.
Herbert M.
Urbassek
, “ Precession of the Earth-Moon system
,”
Eur. J. Phys.
30
, 1427
–1433
(2009
).14.
Olivier
Pujol
,
Christophe
Lagoute
, and
José-Philippe
Pérez
, “ Weight, gravitation, inertia, and tides
,”
Eur. J. Phys.
36
, 065012
(2015
).15.
Chiu-king
Ng
, “ How tidal forces cause ocean tides in the equilibrium
theory
,” Phys. Educ.
50
, 159
–164
(2015
).16.
P. J.
Cregg
, “ Just how much do the planets affect the
tides?
,” Phys. Educ.
52
, 053003
(2017
).17.
Travis
Norsen
,
Mackenzie
Dreese
, and
Christopher
West
, “ The gravitational self-interaction of the Earth's tidal
bulge
,” Am. J. Phys.
85
, 663
–669
(2017
).18.
C. P.
Sonett
,
E. P.
Kvale
,
A.
Zakharian
,
Marjorie A.
Chan
, and
T. M.
Demko
, “ Late Proterozoic and Paleozoic tides, retreat of the
Moon, and rotation of the Earth
,” Science
273
, 100
–104
(1996
).19.
J. O.
Dickey
et al, “ Lunar laser ranging: A continuing
legacy of the Apollo program
,” Science
265
, 482
–490
(1994
).20.
Ian
Garrick-Bethell
,
Viranga
Perera
,
Francis
Nimmo
, and
Maria T.
Zuber
, “ The tidal-rotational shape of the Moon and evidence for
polar wander
,” Nature
512
, 181
–184
(2014
).21.
Chuan
Qin
,
Shijie
Zhong
, and
Roger
Phillips
, “ Formation of the lunar fossil bulges and its implication
for the early Earth and Moon
,” Geophys. Res. Lett.
45
, 1286
–1296
, https://doi.org/10.1002/2017GL076278 (2018
).22.
Immanuel
Kant
, Whether the Earth has Undergone an Alteration of its Axial
Rotation
( Wöchentliche Frag- und
Anzeigungs-Nachricten
, Königsberg
,
1754
).23.
Valeri V.
Makarov
,
Ciprian
Berghea
, and
Michael
Efroimsky
, “ Dynamical evolution and spin-orbit resonances of
potentially habitable exoplanets: The case of GJ 581d
,”
Astrophys. J.
761
, 1
–14
(2012
).24.
J. P.
Zahn
, “ The dynamical tide in close binaries
,”
Astron. Astrophys.
41
, 329
–344
(1975
).25.
J. P.
Zahn
, “ Tidal friction in close binary stars
,”
Astron. Astrophys.
57
, 383
–394
(1977
).26.
J. P.
Zahn
, “ Tidal dissipation in binary systems
,”
EAS Publ. Ser.
29
, 67
–90
(2008
).27.
P.
Auclair-Desrotour
,
J.
Laskar
,
S.
Mathis
, and
A. C. M.
Correia
, “ The rotation of planets hosting atmospheric tides: From
Venus to habitable super-Earths
,” Astron. Astrophys.
603
, A108
(2017
).28.
L. M.
Celnikier
, “ Volcanoes on Io
,” Eur. J.
Phys.
4
, 10
–15
(1983
).29.
Robert H.
Tyler
,
Wade G.
Henning
, and
Christopher W.
Hamilton
, “ Tidal heating in a magma ocean within Jupiter's Moon
Io
,” Astrophys. J. Suppl. Ser.
218
, 1
–17
(2015
).30.
Alar
Toomre
and
Juri
Toomre
, “ Galactic bridges and tails
,”
Astrophys. J.
178
, 623
–666
(1972
).31.
Marc
Fouchard
et al, “ Long-term effects of the Galactic
tide on cometary dynamics
,” Celestial Mech. Dyn.
Astron.
95
, 299
–326
(2006
).32.
C. C.
Counselman
, “ Outcomes of tidal evolution
,”
Astrophys. J.
180
, 307
–314
(1973
).33.
Z.
Kopal
, “ Tidal evolution in close binary systems
,”
Astrophys. Space Sci.
17
, 161
–185
(1972
).34.
W.
van Hamme
, “ On synchronism between axial rotation and orbital motion
in close binary systems
,” Astrophys. Space Sci.
64
, 239
–248
(1979
).35.
36.
See the first paragraph of Section 2 for details
about conservation of angular momentum http://kirkmcd.princeton.edu/examples/spin_orbit.pdf.
37.
J.
Güémez
,
C.
Fiolhais
, and
M.
Fiolhais
, “ The Cartesian diver and the fold
catastrophe
,” Am. J. Phys.
70
, 710
–714
(2002
).38.
Manuel
Fiolhais
and
Rogério
Nogueira
, “ Sistema mecánico con un potencial
catastrófico
,” Rev. Esp. Fís.
34
(1
), 30
–33
(2020
).39.
Bruce G.
Bills
,
Gregory A.
Neumann
,
David E.
Smith
, and
Maria T.
Zuber
, “ Improved estimate of tidal dissipation within Mars from
MOLA observations of the shadow of Phobos
,” J. Geophys.
Res.
110
, E07004
, https://doi.org/10.1029/2004JE002376 (2005
).40.
M.
Efroimsky
and
V.
Lainey
, “ Physics of bodily tides in terrestrial planets and the
appropriate scales of dynamical evolution
,” J. Geophys.
Res.
112
(E12
), E12003
, https://doi.org/10.1029/2007JE002908 (2007
).41.
P.
Hut
, “ Tidal evolution in close binary systems
,”
Astron. Astrophys.
99
, 126
–140
(1981
).42.
Wolfram Research, Inc.
,
Mathematica, Version 11.3
( Wolfram
Research, Inc
., Champaign, IL
,
2018
).43.
G. H.
Darwin
, “ On the bodily tides of viscous and semi-elastic
spheroids, and on the ocean tides upon a yielding nucleus
,”
Philos. Trans. R. Soc.
170
, 1
–35
(1879
).44.
E. C.
Zeeman
, “ Catastrophe theory
,” Sci.
Am.
234
, 65
–83
(1976
).45.
46.
P. T.
Saunders
, An Introduction to Catastrophe Theory
(
Cambridge U. P
., New
York
, 1980
).47.
T.
Poston
and
I.
Stewart
, Catastrophe Theory and Its Applications
( Pitman
, London
,
1978
).© 2020 American Association of Physics Teachers.
2020
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