1) Estimate the distance of Venus from the Sun at its most recent maximum elongation, if the angle of max. elongation was 46 degrees.
The diagram for Problem 1 is shown below, including the angle of maximum elongation:
The diagram for the configuration is shown below:
Θ = 360 (1/P - 1/P') t
where P, P' are the sidereal periods, or P = 365.25 d (Earth) and P' = 687 d (Mars)
Θ = 360 (1/365.25 - 1/687) (36.5) = 16.o8
which is none other than angle E1Sp1.
Now, angle E1p1S is required in order to get the distance - as the information at the bottom of the depiction shows.
Ep1S = 180 - [SE1p1 - E1Sp1]
and SE1p1 = 136 o
Ep1S = 180 - [136 o - 16. o 8 ]= 27. o 2
Again, from plane trigonometry:
sin(p1E1S)/ Sp1 = sin(E1p1S)/ SE1
and, the ratio of planetary distances at time t:
Sp1/SE1 = sin (p1E1S)/ sin (E1p1S)
E1p1S = 27. o2
and p1E1S = SE1p1 = 136 o
Sp1/SE1 = sin (136)/ sin (27.2)
Sp1 = (0.694/ 0.457) a(E) = 1.518 (a(E)) = 1.518 AU