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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.
We neglect the inclination of Mars' orbit:
Seen from the Earth against the background stars the line of sight to Mars is turning retrograde from P_{1}' to P by an angle η which can be be measured. Applying the sine rule to the triangle SE_{1}P_{1} we get: r_{P} / r_{E}
= sin µ / sin(η+β)
and from
(εβ)
+ µ + (η+β) = 180° = ε + µ + ηwe get
r_{P}
/ r_{E} = sin(η+ε)/sin(η+β)Finally the
orbital radius r_{P} = r_{Mars} is given
by:
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. Example: 2007 Dec 03, 23 UT to Dec 24, 23 UT: t = 21 d, η = 7.6° 