Astronomy Principles and Practice Fourth Edition.pdf

Measurement of planetary distances 157
Figure 12.7. Measurement of the distance of a superior planet.
Case (b). The planet is a superior one. In this case the problem is more difficult but can be solved
if the planet’s synodic period S is known.
Let the planet P be in opposition at a given time with the Sun and the Earth in positions S and E
as shown in figure 12.7. The elongation is 180 ◦ .
After t days have elapsed, the Earth’s radius vector SE 1 has moved ahead of the planet’s radius
vector, reducing the elongation from 180 ◦ , at opposition, to the value given by SE 1 P 1 . This value
can, of course, be measured.
In t days, ESP will have increased from 0 ◦ , at opposition, to a value θ,givenby
θ = (n ⊕ − n P )t
where n ⊕ and n P are the angular velocities of Earth and planet in their orbits.
Using relations (12.1), we may write
( 1
θ = 360 − 1 )
t
T ⊕ T P
where T ⊕ and T P are the sidereal periods of Earth and planet respectively.
Then, by equation (12.2),
θ = 360 t S .
Since t and S are both known, θ can be evaluated, that is E 1 SP 1 is calculated. Hence, E 1 P 1 S can
be found from the relation,
E 1 P 1 S = 180 − SE 1 P 1 − E 1 SP 1 .
Using the sine formula of plane trigonometry, we have
sin P 1 E 1 S
SP 1
= sin E 1 P 1 S
SE 1
or
SP 1
= sin P 1 E 1 S
SE 1 sin E 1 P 1 S
giving the distance of the planet from the Sun, again in units of the Earth’s distance from the Sun.