We develop a direct geometric method to determine the orbital parameters and mass of a planet, and we then apply the method to Neptune using high-precision data for the other planets in the solar system. The method is direct in the sense that it does not involve curve fitting. This paper, thereby, offers a new pedagogical approach to orbital mechanics that could be valuable in a physics classroom.

1.
All dates mentioned in this paper are in CE.
2.
U. J.
Le Verrier
, “
Recherches sur les mouvements d'Uranus par UJ Le Verrier
,”
Astron. Nachr.
25
,
53
80
(
1846
).
3.
John Couch
Adams
, “
Explanation of the observed irregularities in the motion of Uranus, on the hypothesis of disturbance by a more distant planet; with a determination of the mass, orbit, and position of the disturbing body
,”
Mon. Not. R. Astron. Soc.
7
,
149
152
(
1846
).
4.
David
Rines
, “
The discovery of the planet Neptune
,”
Pop. Astron.
20
,
482
498
(
1912
); available at http://adsabs.harvard.edu/full/1912PA.....20..482R Bibcode: 1912PA.....20..482R.
5.
W. M.
Smart
, “
John Couch Adams and the discovery of Neptune
,”
Nature
158
(
4019
),
648
652
(
1946
).
6.
Siddharth
Bhatnagar
and
Jayant
Murthy
, “
Game of orbits: A gaming approach to Neptune's discovery
,” arXiv:1807.11280 (
2018
).
7.
Michael Martin
Nieto
,
The Titius-Bode Law of Planetary Distances: Its History and Theory
, 1st ed. (
Elsevier
,
New York
,
2014
).
8.
Ernest W.
Brown
, “
On a criterion for the prediction of an unknown planet
,”
Mon. Not. R. Astron. Soc.
92
,
80
101
(
1931
).
9.
Raymond Arthur
Lyttleton
, “
The rediscovery of Neptune
,”
Vistas Astron.
3
,
25
46
(
1960
).
10.
C. J.
Brookes
, “
On a criterion for the prediction of Neptune
,”
Mon. Not. R. Astron. Soc.
158
,
79
83
(
1972
).
11.
Hon Ming
Lai
,
C. C.
Lam
, and
Kenneth
Young
, “
Perturbation of Uranus by Neptune: A modern perspective
,”
Am. J. Phys.
58
(
10
),
946
953
(
1990
).
12.
Gustav
Eriksson
and
Kevin Garcia
Martin
, “
Discovery of Neptune
,” TRITA-SCI-GRU No. 2018-210 (
KTH, School of Engineering Sciences (SCI)
,
2018
).
13.
J. D.
Giorgini
and
JPL Solar System Dynamics Group
, see <https://ssd.jpl.nasa.gov/horizons.cgi> for “NASA/JPL Horizons On-Line Ephemeris System” (last accessed April 04, 2020).
14.
Inclusion of more objects such as big asteroids like Ceres will nominally increase accuracy.
15.
The short term noise in Figs. 2–4 is removed by taking the moving average.
16.
Herbert
Goldstein
,
Charles P.
Poole
, and
John L.
Safko
,
Classical Mechanics
, 3rd ed. (
Pearson Education
,
London
,
2007
), pp.
92
103
.
17.
Actually, ζ= sign of the radial component of velocity at TI is another unknown parameter while solving for the ellipse. With other parameters fixed, we tried both possibilities (ζ=±1) and have kept ζ=+1 as there is no root analogous to that found in Fig. 5 for ζ=1.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.