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Determining the relative orbital radius of Mars

The opposition of a planet is a chance to determine its approximate relative orbital radius by simple observations.

stationary retrograde
November 15, 2007:

Mars becomes stationary and then starts its retograde opposition loop (-0.9 mag).

December 18, 2007:
Closest approach of Mars and Earth (23:47 UT, 0.5893 AU,
-1.6 mag)
Mars opposition 2007
December 24, 2007:

Mars opposition on Earth (
19:47 UT, 0.5929 AU, -1.6 mag) in constellation Gemini,

39 days after start of retrograde loop.
Mars prograde
January 31, 2008:

Mars becomes stationary to end its retograde opposition loop and proceeds in prograde apparent motion
(-0.6 mag),

38 days after opposition.

We neglect the inclination of Mars' orbit:

Cp. Backhaus

My applet: PlanetMotion

E1, P1
positions of the Earth, and Mars moving retrograde before opposition
E, P
positions of the Earth and Mars at opposition
ε, β central angles of the Earth and Mars due to the motion around the Sun
rE , rP orbital radii of the Earth and the planet

Seen from the Earth against the background stars the line of sight to Mars is turning retrograde from P1' to P by an angle η which can be be measured.

Applying the sine rule to the triangle SE1P1 we get:

rP / rE = sin µ / sin(η+β)
and from
(ε-β) + µ + +β) = 180° = ε + µ + η
we get
rP / rE = sin(η+ε)/sin(η+β)
Finally the orbital radius rP = rMars is given by:
Mars orbital radius
The angles can be computed from the sidereal orbital periods of the Earth (365.25 d) and Mars (686.96 d) and the difference t of time:

ε = t*360°/365.25 d = t*0.986 °/d
β = t*360°/686.96 d = t*0.524 °/d

We have to measure the angle
η in the sky using photos.

2007 Dec 03, 23 UT to Dec 24, 23 UT: t = 21 d, η = 7.6°

ε = 20.7°, β =11.0°

rMars = sin(7.6°+20.7°)/sin(7.6°+11.0°) = 1.49 AU

semimajor axis: 1.524 AU, perihelion 1.405 AU, aphelion 1.639 AU (Mars Fact Sheet, NASA)

2007 Dec 03, 23 UT: rMars/rEarth = 1.5504 AU/0.9857 AU = 1.573

2007 Dec 24, 23 UT: rMars/rEarth = 1.5760 AU/0.9835 AU = 1.602

Web Links
U. Backhaus: Bestimmung der Radien von Planetenbahnen mit Fernglas und Sternkarte (PDF)

U. Backhaus: Die Bewegung der Planeten (PDF)

U. Backhaus: Die Marsbahn (PDF)

W. Mahl: Bestimmung der relativen Entfernung der Planeten Mars und Erde von der Sonne, durch die Beobachtung der Rückläufigkeit von Mars bei seiner Oppositionsstellung

Mars Oppositions (SEDS)

Determining the Orbit of Mars

Close Encounters of the Martian Kind and Other Astronomical Stories of 2003

Updated: 2008, Jan 12