The precession of the perihelion of Mercury’s orbit is calculated using the Laplace–Runge–Lenz vector. An approximate calculation that assumes the orbits of the perturbing planets are circular and coplanar with Mercury’s orbit is within 4.4% of the correct value. A complete calculation that uses the correct elliptical orbit and orientation for each of the perturbing planets is then presented. The precession due to a perturbing planet is proportional to the mass of the planet and is approximately inversely proportional to the cube of its semimajor axis.
REFERENCES
1.
Eric
Doolittle
, “The secular variations of the elements of the orbits of the four inner planets computed for the epoch 1850.0 G.M.T.
,” Trans. Am. Phil. Soc.
22
, 37
–189
(1912
), and references therein.2.
Herbert
Goldstein
, Charles
Poole
, and John
Safko
, Classical Mechanics
(Addison–Wesley
, San Francisco, CA, 2002
), 3rd ed., p. 538
.3.
Jerry B.
Marion
and Stephen T.
Thornton
, Classical Dynamics of Particles and Systems
(Harcourt Brace
, Orlando, FL, 1995
), 4th ed., pp. 318
–321
.4.
Michael P.
Price
and William F.
Rush
, “Nonrelativistic contribution to Mercury’s perihelion precession
,” Am. J. Phys.
47
, 531
–534
(1979
).5.
B.
Davies
, “Elementary theory of perihelion precession
,” Am. J. Phys.
51
, 909
–911
(1983
).6.
Reference 2, pp. 102–106. For more information about the history of the Laplace–Runge–Lenz vector, see
Herbert
Goldstein
, Am. J. Phys.
43
, 737
–738
(1975
);Herbert
Goldstein
, Am. J. Phys.
44
, 1123
–1124
(1976
).7.
See EPAPS Document No. E-AJPIAS-73-011508 for details of the calculations. A direct link to this document may be found in the online article’s HTML reference section. The document may also be reached via the EPAPS homepage ⟨http://www.aip.org/pubservs/epaps.html⟩ or from ftp.aip.org in the directory/epaps. See the EPAPS homepage for more information.
8.
Explanatory Supplement to the Astronomical Almanac
edited by P. Kenneth
Seidelmann
(University Science Books
, Mill Valley, CA, 1992
), p. 697
, Table 15.2.20.9.
See Ref. 8, p.
316
, Table 5.8.1.10.
The sidereal period of Mercury is (Ref. 8, p.
704
, Table 15.6), which gives a conversion factor from radians per revolution to seconds of arc per century of .12.
When measured from the vernal equinox of earth’s orbit along the ecliptic to the ascending node, is called the longitude of the ascending node.
13.
When measured from the ecliptic to the perihelion, is called the argument of the perihelion. The longitude of the perihelion, which is defined as is usually given in astronomical tables instead of .
14.
15.
Reference 1, p.
41
.16.
The normal force, , gives rise to changes in and .
17.
18.
Reference 3, p.
318
.© 2005 American Association of Physics Teachers.
2005
American Association of Physics Teachers
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