A “gravity tunnel” is the name given to a fictitious deep shaft drilled inside the Earth so that objects dropped from the surface of the Earth would free fall without ever touching the walls. It is well known that because of the rotation of the Earth, such tunnels are not straight lines but instead they emerge westward of the antipodal point, when the Earth is approximated as a rotating sphere. In this article, we determine the shape of gravity tunnels by taking into account the polar flattening of the Earth resulting from its rotation. The Earth is described as a McLaurin spheroid, an exact equilibrium shape for a rotating homogeneous deformable body that provides a fair description of the actual shape of the Earth. It turns out that the gravitational force acting on an object located inside the spheroid has a simple form (it is harmonic), so that it is straightforward to compute analytically the free fall trajectories. This study follows a procedure presented several times in this journal and elsewhere, i.e., the trajectory is first computed in the geocentric (non-rotating) frame, and it is then analysed in the terrestrial (rotating) frame. We find that when the flattening of the Earth is taken into account, gravity tunnels have no exit: an object dropped from a point of the surface (other than on the equator or a pole) never reaches the surface again, unless the flattening has very specific (and unnatural) values. We also compute the deviations from the vertical for short falls and compare them to standard eastwards and southwards deviation expressions obtained with other modelizations of the gravity of the rotating Earth, in particular, for a rotating spherical body.

1.
Edwin H.
Hall
, “
Do falling bodies move south? (part I)
,”
Phys. Rev.
7
,
179
190
(
1903
).
2.
Angus
Armitage
, “
The deviation of falling bodies
,”
Ann. Sci.
5
,
342
351
(
1947
).
3.
Harold L.
Burstyn
, “
The deflecting force of the earth's rotation from Galileo to Newton
,”
Ann. Sci.
21
,
47
80
(
1965
).
4.
A. P.
French
, “
The deflection of falling objects
,”
Am. J. Phys.
52
,
199
(
1984
).
5.
John F.
Wild
, “
Simple non-coriolis treatments for explaining terrestrial east-west deflections
,”
Am. J. Phys.
41
,
1057
1059
(
1973
).
6.
Pirooz
Mohazzabi
, “
Free fall and angular momentum
,”
Am. J. Phys.
67
,
1017
1020
(
1999
).
7.
Jacques
Renault
and
Emile
Okal
, “
Investigating the physical nature of the Coriolis effects in the fixed frame
,”
Am. J. Phys.
45
,
631
633
(
1977
).
8.
J. M.
Potgieter
, “
An exact solution for the horizontal deflection of a falling object
,”
Am. J. Phys.
51
,
257
258
(
1983
).
9.
E.
Belorizky
and
J.
Sivardière
, “
Comments on the horizontal deflection of a falling object
,”
Am. J. Phys.
55
,
1103
1104
(
1987
).
10.
Martin S.
Tiersten
and
Harry
Soodak
, “
Dropped objects and other motions relative to the noninertial earth
,”
Am. J. Phys.
68
,
129
138
(
2000
).
11.
A. J.
Simoson
, “
Falling down a hole through the Earth
,”
Math. Mag.
77
,
171
189
(
2004
).
12.
Alexander R.
Klotz
, “
The gravity tunnel in a non-uniform Earth
,”
Am. J. Phys.
83
,
231
237
(
2015
).
13.
Markus
Selmke
, “
A note on the history of gravity tunnels
,”
Am. J. Phys.
86
,
153
(
2018
).
14.
P. W.
Cooper
, “
Through the Earth in forty minutes
,”
Am. J. Phys.
34
,
68
70
(
1966
).
15.
Chandrasekhar
, “
Ellipsoidal figures of equilibrium—An historical account
,”
Com. Pure Appl. Math.
XX
,
251
265
(
1967
).
16.
See, for instance,
John L.
Greenberg
,
The Problem of the Earth's Shape from Newton to Clairaut
(
Cambridge U.P.
,
New York
,
1995
) for an historical account up to the 18th century.
17.
S.
Chandrasekhar
,
Ellipsoidal Figures of Equilibrium
(
Yale U.P.
,
New Haven
,
1969
).
18.
The value adopted for the Krasovskyi spheroid, used in geodesy, is e = 0.0818, see
V. N.
Gan'shin
, Geometria zemnogo ellipsoida, (Moskva : Izd-vo Nedra, 1967); Geometry of the Earth Ellipsoid (English translation), by J. M. Willis, ACIC-TC-1473, Aeronautical Chart and Information Center (St. Louis, 1969).
19.
Isaac
Newton
,
Philosophiae Naturalis Principia Mathematica
, Book III, 3rd ed. (
1726
). English translation by I. Bernard Cohen and Anne Whitman (University of California Press, Oakland, 1999).
20.
Edward A.
Desloge
, “
Horizontal deflection of a falling object
,”
Am. J. Phys.
53
,
581
582
(
1985
).
21.
Forest Ray
Moulton
,
An Introduction to Celestial Mechanics
, 2nd revised ed. (
Dover
,
New York
,
1970
), republication of the original publication of 1914.
See also
J.
Binney
and
S.
Tremaine
,
Galactic Dynamics
, Princeton Series in Astrophysics (
Princeton U.P.
,
Princeton
,
1987
), for a very different derivation of this result.
22.
See Supplementary Material at https://doi.org/10.1119/1.5075716 E-AJPIAS-86-006812-006812 for python programs that can be used to plot figures 5 and 8.

Supplementary Material

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